continuity in an interval

Question

Suppose $f$ is a function as $f:[0,1]\to[0,1]$ and continuous on $[0,1]$. How can I prove that $\exists x_0 \in [0,1]$ such that $f(x_0)= x_0$. Also, how can I prove that $\forall n \in \mathbb {N} \ast \exists a_n \in [0,1] such that {a}{n} = {a}{n}^{n}.

Answer 1

Show that the function $x\mapsto f(x)-x$ is continuous too, is nonpositve at one point, nonegative at one point and therefore zero at some point by the intermediate value theorem.