Independence of choice of faithful representation in reduced $C^*$ crossed product

Question

In the definition of the reduced $C^*$ crossed product associated with an action of a discrete group $G$ on a $C^*$-algebra $A$, one can begin with any faithful representation of $A$ on a Hilbert space (which exists by the GNS construction), go on to represent the twisted group algebra $AG$ on $\ell^2(G)\otimes H$, and then complete. I don't quite understand why this construction is independent of the choice of faithful representation of $A$. I've heard before that it is because such a representation is isometric but I can't quite fill in the details.