8.3: Sequences and Convergence
( \newcommand{\kernel}{\mathrm{null}\,}\)
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:N→X. 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 p∈X and B∈R such that d(p,xn)≤Bfor all n∈N. In other words, the sequence {xn} is bounded whenever the set {xn:n∈N} 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 p∈X, if for every ϵ>0, there exists an M∈N such that d(xn,p)<ϵ for all n≥M. The point p is said to be the limit of {xn}. We write limn→∞xn:=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 |x−y| 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 n≥M1, d(xn,x)<\nicefracϵ2. Similarly we find an M2 such that for all n≥M2 we have d(xn,y)<\nicefracϵ2. Now take an n such that n≥M1 and also n≥M2 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 p∈X if and only if there exists a sequence {an} of real numbers such that d(xn,p)≤anfor all n∈N, and limn→∞an=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 limj→∞xj=(limj→∞xj1,limj→∞xj2,…,limj→∞xjn).
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 j≥M we have d(x,xj)<ϵ. Fix some k=1,2,…,n. For j≥M we have |xk−xjk|=√(xk−xjk)2≤√n∑ℓ=1(xℓ−xjℓ)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 j≥M then |xk−xjk|<\nicefracϵ√n 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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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 x∈X if and only if for every open neighborhood U of x, there exists an M∈N such that for all n≥M we have xn∈U.
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 M∈N such that for all n≥M we have d(x,xn)<ϵ or in other words xn∈B(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 M∈N such that for n≥M we have xn∈U=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, E⊂X a closed set and {xn} a sequence in E that converges to some x∈X. Then x∈E.
Let us prove the contrapositive. Suppose {xn} is a sequence in X that converges to x∈Ec. As Ec is open, says there is an M such that for all n≥M, xn∈Ec. 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 A⊂X. 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 xn∈B(x,\nicefrac1n)∩A. As d(x,xn)<\nicefrac1n, we have that limxn=x.
Exercises
Let (X,d) be a metric space and let A⊂X. Let E be the set of all x∈X 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,|x−y|} 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:N→N 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 K∈N such that for all n≥K we have xn=xK.
A set S⊂X is said to be dense in X if for every x∈X, 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+1⊂Un for all n) sequence of open sets in a metric space (X,d) such that ⋂∞n=1Un={p} for some p∈X. Suppose that {xn} is a sequence of points in X such that xn∈Un. Does {xn} necessarily converge to p? Prove or construct a counterexample.
Let E⊂X be closed and let {xn} be a sequence in X converging to p∈X. Suppose xn∈E for infinitely many n∈N. Show p∈E.
Take R∗={−∞}∪R∪{∞} be the extended reals. Define d(x,y):=|x−y|1+|x−y| if x,y∈R, define d(∞,x):=d(−∞,x)=1 for all x∈R, 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 xn≥n 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+1⊃Vn. Let {xn} be a sequence such that xn∈Vn+1∖Vn and suppose that {xn} converges to p∈X. Show that p∈∂V where V=⋃∞n=1Vn.