problem involving continuity and sequences. Please, help me to solve this problem.
$f\colon[a,b] \rightarrow \mathbb R$ continuous and $ \forall n, \exists x_n \in [a,b]$, such that
 $|f(x_n) - x_n|< \frac{1}{n}$. Prove that $ \exists x \in [a,b]$ such that $f(x)=x$.

We have a sequence $\{x_n\}_1^\infty \subset [a,b]$. By Bolzano-Weirstrass this has a convergent subsequence, say $\{x_{n_k}\} \to x$. $|f(x_{n_k}) - x| \to 0$, so we are done by continuity. 
