Prove that functions $\phi$ that are zero except on some finite subset of $A$ are dense in $\ell^2(A)$
For each finite subset $F \subset A$, define $$ s_F := \sum_{a\in F} |\varphi(a)|^2 $$ Since $\alpha:= \sum_{a\in A} |\varphi(a)|^2 < \infty$, for any $\epsilon > 0, \exists F_0 \subset A$ finite such that $$ |s_F - \alpha| < \epsilon^2 \quad\forall F\supset F_0 \text{ finite} $$ Now define $\psi \in D$ by $$ \psi(a) = \varphi(a) \text{ if } a\in F_0 \text{ and } 0 \text{ otherwise} $$ Then note that $$ d(\varphi,\psi) < \epsilon $$