# Subsequence Convergence

1. Jan 15, 2008

### mrroboto

1. The problem statement, all variables and given/known data

Suppose that {Xn} is a sequence in R. Prove that Xn converges to a if and only if every subsequence of Xn converges to a.

2. Relevant equations

3. The attempt at a solution

Let e>0, choose N in N st n >=N implies |Xn-a| <e. Since a subsequence, nk, is in N and n1<n2<n3...., then nk>=k for all k in N. So, k>= implies |Xnk-a|<e.

That's the first part, but I can't figure out how to start the proof the other way around. i.e. how do you prove that a convergent subsequence implies a convergent sequence?

2. Jan 15, 2008

### EnumaElish

It seems to me the sequence must be Cauchy for it to work.

3. Jan 16, 2008

### morphism

Any sequence is a subsequence of itself. So it's a tautology really.

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