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

(a) Prove that $\displaystyle (x_n)$ converges to a number $\displaystyle A$ iff every subsequence of $\displaystyle (x_n)$ has a subsequence which converges to $\displaystyle A$.

(b) Does $\displaystyle (x_n)$ converge if every subsequence of $\displaystyle (x_n)$ has a subsequence which converges?