I'm trying to show there's a faithful representation on a separable $C^*$-Algebras.
So we introduce the following state $\varphi(a{_n}^*a) = \lVert a_n \rVert^2$, for $a$ in $A$ where $A$ denotes a separable $C^*$-algebra.
For all $n$, we associate $(H_n, \pi_n, x_n)$ the triple given by GNS construction and we set $H = \oplus H_n$.
I already showed that $H$ is separable and now I need to proof the following statement : let be $a \in A$, $\varepsilon < \frac{1}{2} \lVert a \rVert$ and let be $n \in \mathbb{N}$ such that $\lVert a - a_n \rVert < \varepsilon$, show that $\lVert \pi_n(a) \rVert > 0$.
I don't really know where to start, even if I try to build the representation from GNS proof, any idea ?