The Hilbert cube is the space $\displaystyle H = \{\{x_n\}_{n \in \mathbb{N}} \in \ell_\mathbb{R} ^2 : 0 \leq x_k \leq 1/k, \forall k \geq 1\}$ with the metric relative to that of $\displaystyle \ell_\mathbb{R} ^2$.

Let $\displaystyle f : H \longrightarrow \prod _{i=1} ^\infty [0,1]$ given by $\displaystyle f(\{x_n\}) = n\{x_n\}$ and $\displaystyle [0,1]$ with the usual metric.

Prove that f is a homeomorphism.