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

Let [math]\displaystyle{ H=\{(x_1,x_2,...)| \sum x_n^2\lt \infty\} }[/math] and define [math]\displaystyle{ S^{\infty}=\{x\in H| ||x||=1\} }[/math]

Claim

[math]\displaystyle{ S^{\infty} }[/math] is contractible

Proof

A way to see this is via the cellular structure of [math]\displaystyle{ S^{\infty} }[/math]. If [math]\displaystyle{ S^{\infty}=C_0 C_1 ... }[/math] you can always contract [math]\displaystyle{ C_k }[/math] along [math]\displaystyle{ C_{k+1} }[/math] 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?