0708-1300/the unit sphere in a Hilbert space is contractible

Let $H=\{(x_{1},x_{2},...)|\sum x_{n}^{2}<\infty \}$ and define $S^{\infty }=\{x\in H|||x||=1\}$

Claim

$S^{\infty }$ is contractible

Proof

A way to see this is via the cellular structure of $S^{\infty }$. If $S^{\infty }=C_{0}C_{1}...$ you can always contract $C_{k}$ along $C_{k+1}$ like moving contracting the equator along the surface of the earth.

Does this proof only works in separable Hilbert spaces? Is the unit ball in a non-separable Hilbert space contractible?