This is the swindle, isn't it?
There's an elegant way to phrase this with lots of sines and cosines, but working it all out is too much like hard work. Here's the quick and dirty way.
Let T: S∞ → S∞ be the "shift everything down by 1" map.
Then for any point, x and T(x) are not colinear, so the line between them does not go through the origin. We can therefore define a homotopy from the identity on S∞ to T by taking the homotopy tx + (1 - t)T(x) and renormalising so that it is always on the sphere (incidentally, although you are working in ℓ0, by talking about a sphere you implicitly have a norm).
Then we simply contract the image of T, which is a codimension 1 sphere, to a point not on it, say (1,0,0,0,0,...). Again, we can use 'orrible sines and cosines, but renormalising the direct path will do.
(Incidentally, there's nothing special about which space you are taking the sphere in. So long as your space is stable in the sense that X ≅ X ⊕ ℝ then this works)