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

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Homework Statement


Suppose that S is a countably infinite subset of [itex]\ell_2[/itex] with the property that the linear span of S′ is dense in [itex]\ell_2[/itex] whenever S\S′ is finite. Show that there is some S′ whose linear span is dense in [itex]\ell_2[/itex] 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 [itex]\ell_2[/itex] 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
 

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