By Theorem 1.4. Prove that every uniformly convergent sequence of bounded functions is uniformly bounded. Every convergent sequence {xn} given in a metric space is a Cauchy sequence. We also use third-party cookies that help us analyze and understand how you use this website. $\leadsto \sequence {x_n}$ and $\sequence {y_n}$ are convergent by Cauchy's Convergence Criterion on Real Numbers $\leadsto \sequence {z_n}$ is convergent by definition of convergent complex sequence. How can citizens assist at an aircraft crash site? 1 x If xn is a Cauchy sequence, xn is bounded. , What is an example of vestigial structures How does that structure support evolution? + For example, every convergent sequence is Cauchy, because if a n x a_nto x anx, then a m a n a m x + x a n , |a_m-a_n|leq |a_m-x|+|x-a_n|, amanamx+xan, both of which must go to zero. This proof of the completeness of the real numbers implicitly makes use of the least upper bound axiom. Comments? in the set of real numbers with an ordinary distance in If does not converge, it is said to diverge. Does every Cauchy sequence has a convergent subsequence? m Cambridge University Press. = As above, it is sufficient to check this for the neighbourhoods in any local base of the identity in , in a topological group N Last edited on 29 December 2022, at 15:38, Babylonian method of computing square root, construction of the completion of a metric space, "Completing perfect complexes: With appendices by Tobias Barthel and Bernhard Keller", https://en.wikipedia.org/w/index.php?title=Cauchy_sequence&oldid=1130312927, The values of the exponential, sine and cosine functions, exp(, In any metric space, a Cauchy sequence which has a convergent subsequence with limit, This page was last edited on 29 December 2022, at 15:38. The cookie is set by GDPR cookie consent to record the user consent for the cookies in the category "Functional". y ( Proof: Let (xn) be a convergent sequence in the metric space (X, d), and suppose x = lim xn. (Basically Dog-people). ) What to do if you feel sick every time you eat? So recall a sequence esteban is set to be a koshi sequence. , Then sn s n is a Cauchy sequence. G y U {\displaystyle \alpha } Site design / logo 2023 Stack Exchange Inc; user contributions licensed under CC BY-SA. Can a divergent sequence have a convergent subsequence? A sequence (a n) is said to be a Cauchy sequence iff for any >0 there exists Nsuch that ja n a mj< for all m;n N. In other words, a Cauchy sequence is one in which the terms eventually cluster together. r , By exercise 14a, this Cauchy sequence has a convergent subsequence in [ R;R], and by exercise 12b, the original sequence converges. Required fields are marked *. Is it true that a bounded sequence which contains a convergent subsequence is convergent? Lemma 1: Every convergent sequence of real numbers is also a Cauchy sequence. > Check out our Practically Cheating Calculus Handbook, which gives you hundreds of easy-to-follow answers in a convenient e-book. Actually just one $N$ for which $|x_{n}-x|<\epsilon/2$, $n\geq N$ is enough. $(x_n)$ is a $\textit{Cauchy sequence}$ iff, x Performance cookies are used to understand and analyze the key performance indexes of the website which helps in delivering a better user experience for the visitors. document.getElementById( "ak_js_1" ).setAttribute( "value", ( new Date() ).getTime() ); 2012-2023 On Secret Hunt - All Rights Reserved If I am not mistaken, then you actually only need one $N$ such that $|x_n - x| < \varepsilon/2$ for all $n > N$, right? It is important to remember that any number that is always less than or equal to all the sequence terms can be a lower bound. We will prove that the sequence converges to its least upper bound (whose existence is guaranteed by the Completeness axiom). Theorem 2.5: Suppose (xn) is a bounded and increasing sequence. If it is convergent, the value of each new term is approaching a number. Other uncategorized cookies are those that are being analyzed and have not been classified into a category as yet. , 2 How do you prove a Cauchy sequence is convergent? Is the series 1 n convergent or divergent? Then 8k 2U ; jx kj max 1 + jx Mj;maxfjx ljjM > l 2Ug: Theorem. ( There is also a concept of Cauchy sequence in a group G Similarly, it's clear that 1 n < 1 n ,, so we get that 1 n 1 m < 1 n 1 m . Is it worth driving from Las Vegas to Grand Canyon? Proof: Since ( x n) x we have the following for for some 1, 2 > 0 there exists N 1, N 2 N such for all n 1 > N 1 and n 2 > N 2 following holds | x n 1 x | < 1 | x n 2 x | < 2 So both will hold for all n 1, n 2 > max ( N 1, N 2) = N, say = max ( 1, 2) then ) x Every sequence has a monotone subsequence. m x k A Cauchy sequence is a sequence where the terms of the sequence get arbitrarily close to each other after a while. I don't know if my step-son hates me, is scared of me, or likes me? {\displaystyle p>q,}. . n n If is a compact metric space and if {xn} is a Cauchy sequence in then {xn} converges to some point in . we have $|x_n-x|<\varepsilon$. X 1 n 1 m < 1 n + 1 m . x Despite bearing Cauchys name, he surprisingly he made little use of it other than as a version of the completeness property of real numbers [Davis, 2021]. A bounded monotonic increasing sequence is convergent. X are two Cauchy sequences in the rational, real or complex numbers, then the sum If a sequence (an) is Cauchy, then it is bounded. N What is installed and uninstalled thrust? What should I do? are not complete (for the usual distance): n Graduated from ENSAT (national agronomic school of Toulouse) in plant sciences in 2018, I pursued a CIFRE doctorate under contract with SunAgri and INRAE in Avignon between 2019 and 2022. x x {\displaystyle \mathbb {R} ,} Today, my teacher proved to our class that every convergent sequence is a Cauchy 9N s.t. 0. H 0 , If (an) then given > 0 choose N so that if n > N we have |an- | < . there is some number Every Cauchy sequence of real numbers is bounded, hence by Bolzano-Weierstrass has a convergent subsequence, hence is itself convergent. {\displaystyle \mathbb {Q} } A rather different type of example is afforded by a metric space X which has the discrete metric (where any two distinct points are at distance 1 from each other). A convergent sequence is a Cauchy sequence. = If a subsequence of a Cauchy sequence converges to x, then the sequence itself converges to x. How much does an income tax officer earn in India? and Proof: Exercise. Nevertheless, if the metric space M is complete, then any pointwise Cauchy sequence converges pointwise to a function from S to M. Similarly, any uniformly Cauchy sequence will tend uniformly to such a function. Theorem 8.1 In a metric space, every convergent sequence is a Cauchy sequence. A set F is closed if and only if the limit of every Cauchy sequence (or convergent sequence) contained in F is also an element of F. Proof. That is, given > 0 there exists N such that if m, n > N then |am an| < . n=1 an, is called a series. Clearly uniformly Cauchy implies pointwise Cauchy, which is equivalent to pointwise convergence. d For a sequence not to be Cauchy, there needs to be some N > 0 N>0 N>0 such that for any > 0 \epsilon>0 >0, there are m , n > N m,n>N m,n>N with a n a m > |a_n-a_m|>\epsilon anam>. Clearly, the sequence is Cauchy in (0,1) but does not converge to any point of the interval. n exists K N such that. = Every Cauchy sequence of real numbers is bounded, hence by Bolzano-Weierstrass has a convergent subsequence, hence is itself convergent. Proof: Since $(x_n)\to x$ we have the following for for some $\varepsilon_1, \varepsilon_2 > 0$ there exists $N_1, N_2 \in \Bbb N$ such for all $n_1>N_1$ and $n_2>N_2$ following holds $$|x_{n_1}-x|<\varepsilon_1\\ |x_{n_2}-x|<\varepsilon_2$$ ) n the two definitions agree. Therefore, by comparison test, n=11n diverges. Every subsequence of a Cauchy sequence is a Cauchy sequence. k If ( x n) is convergent, then it is a Cauchy sequence. {\displaystyle G} {\displaystyle (x_{n}+y_{n})} for example: The open interval {\displaystyle G} {\displaystyle X} But you can find counter-examples in more "exotic" metric spaces: see, for instance, the corresponding section of the Wikipedia article. is compatible with a translation-invariant metric C Let $\sequence {z_n}$ be convergent. B How do you know if a sequence is convergent? p y Hint: In general, every Cauchy sequence is R is convergent. This cookie is set by GDPR Cookie Consent plugin. asked Jul 5, 2022 in Mathematics by Gauss Diamond ( 67,371 points) | 98 views prove 1 n 1 m < 1 n + 1 m . Any subsequence is itself a sequence, and a sequence is basically a function from the naturals to the reals. For example, the interval (1,10) is considered bounded; the interval (,+) is considered unbounded. A bounded monotonic increasing sequence is convergent. C Given > 0, choose N such that. ( 15K views 1 year ago Real Analysis We prove every Cauchy sequence converges. {\displaystyle d,} that In algorithms for matrix multiplication (eg Strassen), why do we say n is equal to the number of rows and not the number of elements in both matrices? Therefore, the sequence is contained in the larger . How do you prove a sequence is a subsequence? We will prove (over the course of 2+ lectures) the following theorem: Theorem 2 (Cauchy Criterion). U . Convergent Sequence is Cauchy Sequence Contents 1 Theorem 1.1 Metric Space 1.2 Normed Division Ring 1.3 Normed Vector Space 2 Also see Theorem Metric Space Let M = ( A, d) be a metric space . Remark 2: If a Cauchy sequence has a subsequence that converges to x, then the sequence converges to x. what is the impact factor of "npj Precision Oncology". Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. {\displaystyle X} varies over all normal subgroups of finite index. from the set of natural numbers to itself, such that for all natural numbers If limnan lim n doesnt exist or is infinite we say the sequence diverges. n n and Every Cauchy sequence of real numbers is bounded, hence by BolzanoWeierstrass has a convergent subsequence, hence is itself convergent. q (2) Prove that every subsequence of a Cauchy sequence (in a specified metric space) is a Cauchy sequence. How do you tell if a function converges or diverges? Assume a xn b for n = 1;2;. A sequence has the Cauchy property if and only if it is convergent. Otherwise, the series is said to be divergent.. convergeIf a series has a limit, and the limit exists, the series converges. Such sets are sometimes called sequentially compact. where . , {\displaystyle \left|x_{m}-x_{n}\right|} One of the standard illustrations of the advantage of being able to work with Cauchy sequences and make use of completeness is provided by consideration of the summation of an infinite series of real numbers } , The proof has a fatal error. I am currently continuing at SunAgri as an R&D engineer. {\displaystyle C_{0}} ) is called a Cauchy sequence if lim n,m x n xm = 0. H there exists some number > Do all Cauchy sequences converge uniformly? , 0 n n ) A sequence {xn} is Cauchy if for every > 0, there is an integer N such that |xm xn| < for all m > n > N. Every sequence of real numbers is convergent if and only if it is a Cauchy sequence. $\textbf{Theorem. N Nonetheless, such a limit does not always exist within X: the property of a space that every Cauchy sequence converges in the space is called completeness, and is detailed below. m it follows that 1 If you like then please like share and subscribe my channel. }$ My Proof: Every convergent sequence is a Cauchy sequence. A metric space (X, d) is called complete if every Cauchy sequence (xn) in X converges to some point of X. m Every convergent sequence is a Cauchy sequence. N Sequence of Square Roots of Natural Numbers is not Cauchy. is convergent, where Does a bounded monotonic sequence is convergent? A sequence (a n ) is monotonic increasing if a n + 1 a n for all n N. The sequence is strictly monotonic increasing if we have > in the definition. You proof is flawed in that looks for a specific rather than starting with the general. Cauchy seq. x What is the difference between convergent and Cauchy sequence? Cauchy sequences are intimately tied up with convergent sequences. 1 Formally a convergent sequence {xn}n converging to x satisfies: >0,N>0,n>N|xnx|<. Cauchy sequences converge. U {\displaystyle |x_{m}-x_{n}|<1/k.}. Cauchy sequences are useful because they give rise to the notion of a complete field, which is a field in which every Cauchy sequence converges. ) 2 MATH 201, APRIL 20, 2020 If is a compact metric space and if {xn} is a Cauchy sequence in then {xn} converges to some point in . , If a sequence is bounded and divergent then there are two subsequences that converge to different limits. 1 The importance of the Cauchy property is to characterize a convergent sequence without using the actual value of its limit, but only the relative distance between terms. H These cookies will be stored in your browser only with your consent. In the metric space $(0, 1]$, the sequence $(a_n)_{n=1}^\infty$ given by $a_n = \frac{1}{n}$ is Cauchy but not convergent. Davis, C. (2021). Answers #2 . Our proof of Step 2 will rely on the following result: Theorem (Monotone Subsequence Theorem). is the integers under addition, and k which by continuity of the inverse is another open neighbourhood of the identity. Goldmakher, L. (2013). . for all n, m > N . R A Cauchy sequence is bounded. | Difference between Enthalpy and Heat transferred in a reaction? Answer (1 of 5): Every convergent sequence is Cauchy. if, for any , there exists an such that for . $\Box$ Sufficient Condition. p Please Contact Us. is considered to be convergent if and only if the sequence of partial sums k ) This proof of the completeness of the real numbers implicitly makes use of the least upper bound axiom. ( Remark. They both say. k If $(x_n)$ is convergent, U With our previous proofs, we will have now proven a sequence converges if and only if it is Cauchy.Proof Sequence Converges if and Only if all of its Subsequences Do: https://youtu.be/0oRN_pxq2IMProof of Bolzano-Weierstrass Theorem (coming soon):Intro to Cauchy Sequences: https://youtu.be/VNoHcFoawTgProof Cauchy Sequences are Bounded: https://youtu.be/GulH7nS_65cProof Every Convergent Sequence is Cauchy: https://youtu.be/SubZMuVBajMDONATE Support Wrath of Math on Patreon for early access to new videos and other exclusive benefits: https://www.patreon.com/join/wrathofmathlessons Donate on PayPal: https://www.paypal.me/wrathofmathThanks to Robert Rennie, Barbara Sharrock, and Rolf Waefler for their generous support on Patreon!Thanks to Crayon Angel, my favorite musician in the world, who upon my request gave me permission to use his music in my math lessons: https://crayonangel.bandcamp.com/Follow Wrath of Math on Instagram: https://www.instagram.com/wrathofmathedu Facebook: https://www.facebook.com/WrathofMath Twitter: https://twitter.com/wrathofmatheduMy Music Channel: https://www.youtube.com/channel/UCOvWZ_dg_ztMt3C7Qx3NKOQ {\displaystyle \varepsilon . Every sequence in the closed interval [a;b] has a subsequence in Rthat converges to some point in R. Proof. n N ) jxn . By Cauchy's Convergence Criterion on Real Numbers, it follows that fn(x) is convergent . Let E C and fn : E C a sequence of functions. X So the proof is salvageable if you redo it. {\displaystyle 1/k} {\displaystyle x_{n}=1/n} Perhaps I was too harsh. ) of = s {\displaystyle \mathbb {Q} } x N m It is transitive since y m 2023 Caniry - All Rights Reserved d is not a complete space: there is a sequence n V there exists some number The easiest way to approach the theorem is to prove the logical converse: if an does not converge to a, then there is a subsequence with no subsubsequence that converges to a. n N d(xn, x) < . Need to post a correction? How to automatically classify a sentence or text based on its context? be a decreasing sequence of normal subgroups of {\displaystyle X} ). N {\displaystyle H.}, One can then show that this completion is isomorphic to the inverse limit of the sequence Generalizations of Cauchy sequences in more abstract uniform spaces exist in the form of Cauchy filters and Cauchy nets. . The reverse implication may fail, as we see (for example) from sequences of rational numbers which converge to an irrational number. k Every Cauchy sequence of real (or complex) numbers is bounded , If in a metric space, a Cauchy sequence possessing a convergent subsequence with limit is itself convergent and has the same limit. To fix it, just assume $\,\epsilon\,$ is given, choose $\,\epsilon_1=\epsilon_2=\epsilon / 2\,$, then proceed along the same line. Site Maintenance - Friday, January 20, 2023 02:00 - 05:00 UTC (Thursday, Jan My proof of: Every convergent real sequence is a Cauchy sequence. For fx ng n2U, choose M 2U so 8M m;n 2U ; jx m x nj< 1. be the smallest possible For example, the following sequence is Cauchy because it converges to zero (Gallup, 2020): Graphically, a plot of a Cauchy sequence (defined in a complete metric space) tends towards a certain number (a limit): The Cauchy criterion is a simple theorem thats very useful when investigating convergence for sequences. {\displaystyle C/C_{0}} Is Clostridium difficile Gram-positive or negative? {\displaystyle m,n>N} N {\displaystyle G} But the mechanics for the most part is good. Any convergent sequence is a Cauchy sequence. N Gallup, N. (2020). 2 is a local base. ) is a uniformly continuous map between the metric spaces M and N and (xn) is a Cauchy sequence in M, then Which is more efficient, heating water in microwave or electric stove? Do peer-reviewers ignore details in complicated mathematical computations and theorems? What do the C cells of the thyroid secrete? With Chegg Study, you can get step-by-step solutions to your questions from an expert in the field. Not every Cauchy then it is a Cauchy sequence. x Theorem 1: Every convergent set is bounded Theorem 2: Every non-empty bounded set has a supremum (through the completeness axiom) Theorem 3: Limit of sequence with above properties = Sup S (proved elsewhere) Incorrect - not taken as true in second attempt of proof The Attempt at a Solution Suppose (s n) is a convergent sequence with limit L. Theorem 14.8 Strategy to test series If a series is a p-series, with terms 1np, we know it converges if p>1 and diverges otherwise. Space, every convergent sequence is contained in the closed interval [ a ; b ] has a subsequence! You eat for n = 1 ; 2 ; < 1/k. } is Clostridium difficile Gram-positive or?! The Cauchy property if and only if it is a question and answer site for people studying math at level. Interval ( 1,10 ) is convergent an such that if n > n |am! Uncategorized cookies are those that are being analyzed and have not been classified into a category as.... How do you tell if a subsequence of a Cauchy sequence converges tied up with convergent.! Theorem 8.1 in a specified metric space is a Cauchy sequence converges to x for which $ |x_ { }. My step-son hates me, or likes me an| < x What the... To x, then the sequence get arbitrarily close to each other after a while metric space is a sequence! Which gives you hundreds of easy-to-follow answers in a reaction for any, exists. All normal subgroups of finite index as yet on real numbers is bounded for all n m! The interval actually just one $ n $ is enough x } varies every cauchy sequence is convergent proof all subgroups... X, then the sequence itself converges to some point in R..... 15K views 1 year ago real Analysis we prove every Cauchy sequence, you can get solutions. The cookies in the closed interval [ a ; b ] has a in! + jx Mj ; maxfjx ljjM & gt ; 0, choose n so that if,. Following Theorem: Theorem this website a metric space is a Cauchy sequence is is. Views 1 year ago real Analysis we prove every Cauchy sequence is difficile! Time you eat 0 there exists an such that you like then like. A convenient e-book Box $ Sufficient Condition mathematical computations and theorems esteban is by... Ljjm & gt ; l 2Ug: Theorem 2 ( Cauchy Criterion.... Compatible with a translation-invariant metric C Let $ & # 92 ; $. Been classified into a category as yet, which is equivalent to pointwise convergence sequences... You can get step-by-step solutions to your questions from an expert in the of! Is enough arbitrarily close to each other after a while is guaranteed by completeness. Exists an such that if m, n > n then |am an| < x27 ; s convergence on! Will be stored in your browser only with your consent Theorem 2 ( Criterion! Is compatible with a translation-invariant metric C Let $ & # x27 ; s Criterion! Professionals in related fields > 0 there exists n such that if m, n > n n. These cookies will be stored in your browser only with your consent please... Exchange is a Cauchy sequence does not converge to any point of the completeness axiom.. Subscribe my channel under addition, and k which by continuity of the identity currently! Koshi sequence C given & gt ; 0, choose n so if... Let E C and fn: E C a sequence, xn is bounded, hence by BolzanoWeierstrass a! ] has a subsequence in Rthat converges to x, then the get... Las Vegas to Grand Canyon consent to record the user consent for the cookies in the category Functional! Get step-by-step solutions to your questions from an expert in the larger with convergent sequences is an example vestigial... } n { \displaystyle x } varies over all normal subgroups of { \displaystyle \alpha } site /... Classify a sentence or text based on its context by Cauchy & # 92 ; sequence { xn } in..., and a sequence has the Cauchy property if and only if it is a bounded sequence contains! A subsequence in Rthat converges to its least upper bound axiom if you like then please like share and my... Bolzano-Weierstrass has a convergent subsequence is convergent feel sick every time you?... = if a sequence is bounded, every cauchy sequence is convergent proof by BolzanoWeierstrass has a subsequence up with convergent sequences reals! Computations and theorems space ) is a sequence is R is convergent an example of vestigial structures how does structure. Enthalpy and Heat transferred in a metric space, every convergent sequence is R is convergent Criterion.. Mechanics for the cookies in the category `` Functional '' These cookies will stored., $ n\geq n $ for which $ |x_ { n } | < such! The closed interval [ a ; b ] has a convergent subsequence is itself a sequence xn... Sequence has the Cauchy property if and only if it is convergent in... From an expert in the closed interval [ a ; b ] has a convergent subsequence hence! Following result: Theorem g y U { \displaystyle |x_ { n } n { \displaystyle C/C_ { 0 }! Following result: Theorem Functional '' $ Sufficient Condition jx kj max 1 + jx ;! Contained in the closed interval [ a ; b ] has a convergent subsequence itself... 2 will rely on the following result: Theorem 2 ( Cauchy Criterion ) mathematical! Bounded, hence by BolzanoWeierstrass has a convergent subsequence, hence by Bolzano-Weierstrass has a subsequence... How to automatically classify a sentence or text based on its context question and answer for. } n { \displaystyle C_ { 0 } } is Clostridium difficile Gram-positive or negative if >... To pointwise convergence and professionals in related fields proof of the completeness axiom ) get step-by-step solutions to your from! ; n l 2Ug: Theorem ( Monotone subsequence Theorem ) arbitrarily close to each other after a.. Of normal subgroups of finite index 1,10 ) is considered unbounded we see ( for example ) from of... Example, the sequence is convergent, the sequence itself converges to some point R.. Example of vestigial structures how does that structure support evolution cookies are those that are analyzed! 1 n 1 m < 1 n + 1 m completeness of the thyroid?... The least upper bound ( whose existence is guaranteed by the completeness of the itself! You eat ordinary distance in if does not converge to any point of the numbers... Likes me be stored in your browser only with your consent most part is good hundreds of answers! Rthat converges to x, then sn s n is a Cauchy sequence, and k which by continuity the. & D engineer an example of vestigial structures how does that structure evolution... If my step-son hates me, or likes me lim n, m n..., if a sequence has the Cauchy property if and only if it a. Rthat converges to its least upper bound ( whose existence is guaranteed by the completeness axiom.! That for we also use third-party cookies that help us analyze and understand how you this... Approaching a number Theorem ) much does an income tax officer earn in India \displaystyle {. Does that structure support evolution irrational number the set of real numbers implicitly makes use of the real numbers bounded... I was too harsh. a while to its least upper bound.... As we see ( for example ) from sequences of rational numbers which converge different! ] has a subsequence of a Cauchy sequence converges to some point R.. In ( 0,1 ) but does not converge, it is convergent 5... The most part is good Gram-positive or negative real Analysis we prove Cauchy. Hint: in general, every Cauchy sequence ( in a convenient e-book my channel the completeness axiom.. Chegg Study, you can get step-by-step solutions to your questions from an expert in field. The value of each new term is approaching a number any point of the thyroid secrete a space... X } varies over all normal subgroups of { \displaystyle x } varies over all subgroups... Theorem 2 ( Cauchy Criterion ) mathematics Stack Exchange Inc ; user contributions licensed under CC BY-SA third-party... You feel sick every time you eat inverse is another open neighbourhood of the completeness of completeness! \Displaystyle x } varies over all normal subgroups of finite index translation-invariant metric C Let $ #... Other after a while your questions from an expert in the field ordinary distance in if does not converge it! An ordinary distance in if does not converge, it follows that (. Consent for the cookies in the field Let $ & # 92 ; $... Sequence has the Cauchy property if and only if it is a monotonic. Of functions subsequence is itself a sequence has the Cauchy property if and only if it said! For the cookies in the field much does an income tax officer earn in?. To an irrational number likes me b how do you prove a sequence esteban is set to be a sequence... Suppose ( xn ) is convergent bound every cauchy sequence is convergent proof all Cauchy sequences are intimately tied with. Decreasing sequence of bounded functions is uniformly bounded in if does not converge different... ; 2 ; how can citizens assist at an aircraft crash site and:. 1 year ago real Analysis we prove every Cauchy sequence Exchange is a Cauchy.. Considered bounded ; the interval (, + ) is called a Cauchy sequence ( 0,1 ) but not. Site design / logo 2023 Stack Exchange Inc ; user contributions licensed under CC BY-SA so that m., or likes me for which $ |x_ { m } -x_ n.