Every subsequence has a subsequence that converges to x

Related BrainMass Solutions

• Prove Compactness of a Closed Subset

  Theorem 2: A subset of S of a metric space X is compact if, and only if, every sequence in S has a subsequence that converges to a point in S. Proof: We consider a closed subset S in a compact metric space C.

• Prove that if A is a family of functions in C_0 such that A is uniformly bounded and equicontinuous, then every sequence of functions from A has a convergent subsequence in the sup-norm.

  This solution is comprised of a detailed explanation to prove that if A is a family of functions in C_0 such that A is uniformly bounded and equicontinuous, then every sequence of functions from A has a convergent subsequence in the sup-norm.

• Compact Subset of R^m with Convergent Sequences

  From the condition, each subsequence of (x_n) converges to x, then I claim that (x_n) also converges to x.

• Real Analysis : Converging Sequences

  26066 Real Analysis : Converging Sequences Assume (a_n) is a bounded sequence with the property that every convergent subsequence of (a_n) converges to the same limit a belong to R.show that (a_n) must converges to a.

• Converging Subsequences

  156007 Converging Subsequence Theorem: Suppose that a sequence S of real numbers has a subsequence that converges to a real number a. Then S converges to a.

• Euclidean metric proof

  So it suffices to prove that (please see the attached file) has a convergent subsequence. From the Bolzano-Weierstrass theorem, every bounded sequence in has a convergent subsequence.

• Characterize Real Numbers

  Compact means that every sequence in R has a convergent subsequence. But we consider N={1,2,3,...}, the set of all natural numbers. N is a sequence in R, but for any subsequence of N, we have d(x,y)=1 for any x<>y.

• Convergent Subsequence and Limit of Sequence

  The expert states a tangible reason why your subsequence converges to the claimed limits.

• monotone sequence cluster points

  Definition: A real number x is called a cluster point of the sequence provided some subsequence of the sequence converges to x. Prove that a monotone sequence can have at most one cluster point.