# Math Help - sequence, convergence

1. ## sequence, convergence

Let $(x_n)$ be a sequence of real numbers.

(a) Prove that $(x_n)$ converges to a number $A$ iff every subsequence of $(x_n)$ has a subsequence which converges to $A$.
(b) Does $(x_n)$ converge if every subsequence of $(x_n)$ has a subsequence which converges?

2. I figured out part b, but part a is confusing me.

3. Originally Posted by poincare4223
(a) Prove that $(x_n)$ converges to a number $A$ iff every subsequence of $(x_n)$ has a subsequence which converges to $A$.
If $\{x_n\}$ is convergent then all subsequences converge to the same limit. Thus, the forward direction follows.

To prove the backwards direction assume that $\{x_n\}$ [u]did not[/b] converge to $A$. Then it means there is some $\epsilon > 0$ so that for any $N>0$ we have $|x_n - A| \geq \epsilon$ for some $n>N$. Thus, we can form a subsequence, $\{x_{n_k}\}$ with $|x_{n_k} - A|\geq \epsilon$. But this subsequence cannot possible have a subsequence converging to $A$ . Contradiction.