Hilbert Sequence Space is Homeomorphic to Countable Infinite Product of Real Number Spaces

Theorem
Let $A$ be the set of all real sequences $\left\langle{x_i}\right\rangle$ such that the series $\displaystyle \sum_{i \mathop \ge 0} x_i^2$ is convergent.

Let $\ell^2 = \left({A, d_2}\right)$ be the Hilbert sequence space on $\R$.

Let $\left({\R, \tau_d}\right)$ denote the real number line under the Euclidean topology.

Let $\R^\omega = \displaystyle \prod_{i \mathop \in \N} \left({\R, \tau_d}\right)$ denote the countable-dimensional real Cartesian space under the product metric.

Then $\ell^2$ is homeomorphic to $\R^\omega$.