The Hilbert cube homeomorphic to the cube?

The Hilbert cube is the space $H = \{\{x_n\}_{n \in \mathbb{N}} \in \ell_\mathbb{R} ^2 : 0 \leq x_k \leq 1/k, \forall k \geq 1\}$ with the metric relative to that of $\ell_\mathbb{R} ^2$ .
Let $f : H \longrightarrow \prod _{i=1} ^\infty [0,1]$ given by $f(\{x_n\}) = n\{x_n\}$ and $[0,1]$ with the usual metric.
Prove that f is a homeomorphism.