Prove set of sequences is a basis

Let c_00 be the subspace of all sequences of complex numbers that are "eventually zero". i.e. for an element x∈c_00, ∃N∈N such that xn=0,∀n≥n.



Let {e_i}, i∈N be the set where e_i is the sequence in c_00 given by (e_i)_n =1 if n=i and (e_i)_n=0 if n≠i.



Show that (e_i), i∈N is a basis for c_00.



So I need to show it's linearly independent and that it spans c_00. I am not sure how to go about proving this makes it confusing is that it's an infinite set, so I can't use the usual method and take a finite number of vectors.



I have an idea of how to prove linear independence, but not spanning.



Any tips/hints?



Thanks

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