# Subsequences and convergence

1. Oct 11, 2008

### JinM

1. The problem statement, all variables and given/known data
I had this question on my midterm yesterday:

Let s_n be a convergent sequence. Let lim s_n = s. Show, using the definition of convergence, that all subsequences of s_n converge to the same limit.

3. The attempt at a solution

Since s_n converges, then for any e > 0, we can find N such that |s_n - s| < e for all n > N.

Thus s - e < s_n < s + e. Also, define S = {s_(n_k) | k \in N and n > N}. Then any element s_n_k for which n > N belongs to S, which is a subset of {s_n | n > N}, which is again a subset of the interval [-(s + e), s + e].

For the sake of completeness, there are only finitely many s_n_k's for which n < N. Let N_0 be the greatest n for which s_n_k does not belong to S. Then for all n > N_0, s_n_k belongs to set S. So we just have to pick our N to be greater than this N_0.

This shows that |s_n_k - s| < e, and we are done. Is this enough, or have I missed anything?