Let $X$ be a locally compact, separable metric space and $\mu_n$ a sequence of probability measures on $S$. Let $\mathfrak{C}$ be a convergence determining class for the weak topology (for instance, assume that $\mathfrak{C} = C_b(X)$, i.e. all bounded continuous functions on $X$).

Then, if $\mu_n(f), n \geq 1$ is Cauchy for all $f \in \mathfrak{C}$, does it follow that $\mu_n$ converges weakly?

Alternatively, if $C = C_b(X)$ then writing $m(f) = \lim_n \mu_n(f)$ does the Riesz Representation theorem apply to $m(f)$? In general, we need some tightness conditions on a functional $m(f)$ to apply RRT. However, given the particular form of the functional, is there a way to detect whether the sequence "loses mass at infinity"?

If not, is there a constructive counter-example of this?

Edit: OP here. Thanks for all the suggestions so far. To clarify something: I'm well aware of tightness of the sequence measures being equivalent to precompactness (i.e. Prokhorov's Theorem) which then through sequential compactness would imply a weak limit of $\mu_n$. The issue here is (or was) how to prove that the sequence is uniformly tight in the first place, given only that the integrals against functions $\mu_n(f)$ themselves converge.