Dense subspace in $l^{2}$

Question

Let $(\lambda_{n})_{n=1}^{\infty}$ be a sequence of scalars nonzero, $S = \{x = (\xi_{j}) \in l^{2}: \sum^{\infty}_{j=1}|\lambda_{j}||\xi_{j}|<\infty\}$ and the opetator $T:S \rightarrow l^{2}$ defined by $Tx=(\lambda_{j}\xi_{j})$. Show that $S$ and $R(T)$ are both dense in $l^{2}$.

Answer 1

Hint: Show that the following subspace is dense in $l^2$.

$$ F:= \{ (x_n)\in l^2 \ : \ \text{all but finitely many } x_j's \text{ are zero} \}.$$ Then show $F\subset S$ and $F\subset R(T)$ and conclude.