Does Every Subsequence of a Sequence Converge to the Same Limit?

[mrroboto]

Homework Statement

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.

Homework Equations

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?

 

[EnumaElish]

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

 

[morphism]

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