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8.3: Sequences and Convergence

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Sequences

The notion of a sequence in a metric space is very similar to a sequence of real numbers.

A sequence in a metric space (X,d) is a function x:NX. As before we write xn for the nth element in the sequence and use the notation {xn}, or more precisely {xn}n=1.

A sequence {xn} is bounded if there exists a point pX and BR such that d(p,xn)Bfor all nN. In other words, the sequence {xn} is bounded whenever the set {xn:nN} is bounded.

If {nj}j=1 is a sequence of natural numbers such that nj+1>nj for all j then the sequence {xnj}j=1 is said to be a subsequence of {xn}.

Similarly we also define convergence. Again, we will be cheating a little bit and we will use the definite article in front of the word limit before we prove that the limit is unique.

A sequence {xn} in a metric space (X,d) is said to converge to a point pX, if for every ϵ>0, there exists an MN such that d(xn,p)<ϵ for all nM. The point p is said to be the limit of {xn}. We write limnxn:=p.

A sequence that converges is said to be convergent. Otherwise, the sequence is said to be divergent.

Let us prove that the limit is unique. Note that the proof is almost identical to the proof of the same fact for sequences of real numbers. In fact many results we know for sequences of real numbers can be proved in the more general settings of metric spaces. We must replace |xy| with d(x,y) in the proofs and apply the triangle inequality correctly.

[prop:mslimisunique] A convergent sequence in a metric space has a unique limit.

Suppose that the sequence {xn} has the limit x and the limit y. Take an arbitrary ϵ>0. From the definition we find an M1 such that for all nM1, d(xn,x)<\nicefracϵ2. Similarly we find an M2 such that for all nM2 we have d(xn,y)<\nicefracϵ2. Now take an n such that nM1 and also nM2 d(y,x)d(y,xn)+d(xn,x)<ϵ2+ϵ2=ϵ. As d(y,x)<ϵ for all ϵ>0, then d(x,y)=0 and y=x. Hence the limit (if it exists) is unique.

The proofs of the following propositions are left as exercises.

[prop:msconvbound] A convergent sequence in a metric space is bounded.

[prop:msconvifa] A sequence {xn} in a metric space (X,d) converges to pX if and only if there exists a sequence {an} of real numbers such that d(xn,p)anfor all nN, and limnan=0.

Convergence in euclidean space

It is useful to note what convergence means in the euclidean space Rn.

[prop:msconveuc] Let {xj}j=1 be a sequence in Rn, where we write xj=(xj1,xj2,,xjn)Rn. Then {xj}j=1 converges if and only if {xjk}j=1 converges for every k, in which case limjxj=(limjxj1,limjxj2,,limjxjn).

For R=R1 the result is immediate. So let n>1.

Let {xj}j=1 be a convergent sequence in Rn, where we write xj=(xj1,xj2,,xjn)Rn. Let x=(x1,x2,,xn)Rn be the limit. Given ϵ>0, there exists an M such that for all jM we have d(x,xj)<ϵ. Fix some k=1,2,,n. For jM we have |xkxjk|=(xkxjk)2n=1(xxj)2=d(x,xj)<ϵ. Hence the sequence {xjk}j=1 converges to xk.

For the other direction suppose that {xjk}j=1 converges to xk for every k=1,2,,n. Hence, given ϵ>0, pick an M, such that if jM then |xkxjk|<\nicefracϵn for all k=1,2,,n. Then \[d(x,x^j) = \sqrt{\sum_{k=1}^n {\bigl(x_k-x_k^j\bigr)}^2} < \sqrt{\sum_{k=1}^n {\left(\frac{\epsilon}{\sqrt{n}}\right)}^2} = \sqrt{\sum_{k=1}^n \frac

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{n}} = \epsilon .\] The sequence {xj} converges to xRn and we are done.

Convergence and topology

The topology, that is, the set of open sets of a space encodes which sequences converge.

[prop:msconvtopo] Let (X,d) be a metric space and {xn} a sequence in X. Then {xn} converges to xX if and only if for every open neighborhood U of x, there exists an MN such that for all nM we have xnU.

First suppose that {xn} converges. Let U be an open neighborhood of x, then there exists an ϵ>0 such that B(x,ϵ)U. As the sequence converges, find an MN such that for all nM we have d(x,xn)<ϵ or in other words xnB(x,ϵ)U.

Let us prove the other direction. Given ϵ>0 let U:=B(x,ϵ) be the neighborhood of x. Then there is an MN such that for nM we have xnU=B(x,ϵ) or in other words, d(x,xn)<ϵ.

A set is closed when it contains the limits of its convergent sequences.

[prop:msclosedlim] Let (X,d) be a metric space, EX a closed set and {xn} a sequence in E that converges to some xX. Then xE.

Let us prove the contrapositive. Suppose {xn} is a sequence in X that converges to xEc. As Ec is open, says there is an M such that for all nM, xnEc. So {xn} is not a sequence in E.

When we take a closure of a set A, we really throw in precisely those points that are limits of sequences in A.

[prop:msclosureapprseq] Let (X,d) be a metric space and AX. If x¯A, then there exists a sequence {xn} of elements in A such that limxn=x.

Let x¯A. We know by that given \nicefrac1n, there exists a point xnB(x,\nicefrac1n)A. As d(x,xn)<\nicefrac1n, we have that limxn=x.

Exercises

Let (X,d) be a metric space and let AX. Let E be the set of all xX such that there exists a sequence {xn} in A that converges to x. Show that E=¯A.

a) Show that d(x,y):=min{1,|xy|} defines a metric on R. b) Show that a sequence converges in (R,d) if and only if it converges in the standard metric. c) Find a bounded sequence in (R,d) that contains no convergent subsequence.

Prove

Prove

Suppose that {xn}n=1 converges to x. Suppose that f:NN is a one-to-one and onto function. Show that {xf(n)}n=1 converges to x.

If (X,d) is a metric space where d is the discrete metric. Suppose that {xn} is a convergent sequence in X. Show that there exists a KN such that for all nK we have xn=xK.

A set SX is said to be dense in X if for every xX, there exists a sequence {xn} in S that converges to x. Prove that Rn contains a countable dense subset.

Suppose {Un}n=1 be a decreasing (Un+1Un for all n) sequence of open sets in a metric space (X,d) such that n=1Un={p} for some pX. Suppose that {xn} is a sequence of points in X such that xnUn. Does {xn} necessarily converge to p? Prove or construct a counterexample.

Let EX be closed and let {xn} be a sequence in X converging to pX. Suppose xnE for infinitely many nN. Show pE.

Take R={}R{} be the extended reals. Define d(x,y):=|xy|1+|xy| if x,yR, define d(,x):=d(,x)=1 for all xR, and let d(,):=2. a) Show that (R,d) is a metric space. b) Suppose that {xn} is a sequence of real numbers such that for xnn for all n. Show that limxn= in (R,d).

Suppose that {Vn}n=1 is a collection of open sets in (X,d) such that Vn+1Vn. Let {xn} be a sequence such that xnVn+1Vn and suppose that {xn} converges to pX. Show that pV where V=n=1Vn.

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This page titled 8.3: Sequences and Convergence is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Jiří Lebl via source content that was edited to the style and standards of the LibreTexts platform.

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