"Refinement" of the existence of faithful representations of C*-algebras?

Question

Conway, in a course in operator theory, brings the statement 1. below as a theorem and statement 2. below as an exercise. Still, he states that 2. refines 1., but I can't see it.

1. Every C*-algebra $\mathcal{A}$ has a faithful representation. $A$ has a faithful representation $(\pi,\mathcal{H})$ with $\mathcal{H}$ separable if and only if there are a countable number of states on $\mathcal{A}$ that separate the points of $\mathcal{A}_+$, each of which defines a separable representation. In particular, every separable C*-algebra has a faithful, separable representation.
2. If $\mathcal{A}$ is a separable C*-algebra and $\left\{\phi_n\right\}$ a weak* dense sequence in the state space, put $\phi=\sum_n2^{-n}\phi_n$. Then $\phi$ is a state and the representation $\pi_{\phi}$ is faithful.

How does 2. refine 1.?

Answer 1

Branimir had posted an answer as a comment:

2. refines 1. because the representation constructed in 1. is, in general, a direct sum of infinitely many GNS representations, i.e., representations constructed from states on $\mathcal A$, whilst 2. gives you a single GNS representation that does the job.