The Schauder Fixed Point Theorem

Let $E = \left\{ x \in l^2(\mathbb {N}) : \left\| x\right\| \leq 1 \right\}$ and for $x$ in $E$ define $f(x) = \left( (1 - \left\| x\right\|^2)^{1/2}, x(1), x(2), \dots \right)$. Show that $f(E) \subseteq E$, $f$ is continuous, and $f$ has no fixed points.