How to prove this element is strictly positive?

Question

Let $A$ be a $C^*\text{-algebra}$ and $A_+$ denote the positive elements. An element $a\in A_+$ is called strictly positive if $\overline{aAa}=A$.
Want to prove: if $(e_n)$ is an approximate identity of $A$ , then $a:=\sum_{n=1}^{\infty}\frac{1}{2^n}e_n$is strictly positive. i.e, I want to prove that if $b\in A$, then I can find a sequence $x_n\in A$ such that $b=\lim(ax_na)$. By an approximate identity of $A$ we mean that : $(e_n)$ is increasing, positive,$\|e_n\|<1$ and for all $a\in A$ : $\|(ae_n)-a\|$ converges to $0$ .
Any help or suggestion is really appreciated. Thank you!