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.