strictly positive element iff A contains a countable approximative unit

Question

I search a proof of: Let A be a c$^*$-algebra and let $(u_n)_{n\in\mathbb{N}}$ an approximative unit in A. Then $a=\sum\limits_{n=1}^{\infty}\frac{u_n}{2^n}$ is strictly positive. Could anybody tell me how to prove it? The converse is true too and the converse i have proved. Regards

Edit: $a\in A$ is strictly positive, if for every state $\eta$ of A: $\eta(a)>0$.
I only found this strictly positive elements in $C^*$-algebra but this answer uses an other definition.

Answer 1

If $\phi(a)=0$ for some state $\phi$, then $\phi(u_n)=0$ for all $n$, because $0\leq\phi(u_n)\leq2^n \phi(a)$.

Now, for any $b\in A_+$, $$ \phi(b)=\lim_n \phi(u_nbu_n)=\lim_n\phi(u_n^{1/2}[u_n^{1/2}bu_n^{1/2}]u_n^{1/2})\leq\limsup_n\|u_n^{1/2}bu_n^{1/2}\|\,\phi(u_n) \\ \leq\|b\|\limsup_n\phi(u_n)=0. $$ So $\phi(b)=0$ for all $b\geq0$, and thus $\phi=0$ as $A_+$ spans $A$.

So, for any state $\phi\ne0$, we have $\phi(a)>0$.