Constructing a subset of l_2 with dense linear span with finite complement

Mathmos6

Homework Statement

Suppose that S is a countably infinite subset of $\ell_2$ with the property that the linear span of S′ is dense in $\ell_2$ whenever S\S′ is finite. Show that there is some S′ whose linear span is dense in $\ell_2$ and for which S\S′ is infinite.

The Attempt at a Solution

I have tried repeatedly to solve this by constructing a series of subsets of some arbitrary S, such that the complement is finite and of increasing size. I haven't actually used the fact that we're working in $\ell_2$ here, so it's quite likely that I'm meant to use some property of Hilbert spaces - however, I'm not sure what. Could anyone please help? Thankyou very much; Mathmos6