= Solution
For a bounded measurable $f$, the dual estimate for <total variation distance> gives
$$
\left|\int f\,d\mu_n-\int f\,d\mu\right|
\leq2\lVert f\rVert_\infty
\sup_A|\mu_n(A)-\mu(A)|.
$$
The right-hand side tends to zero, so in particular the integrals converge for every bounded continuous $f$. Thus <weak convergence of random variables> follows.
The converse fails. On $\mathbb R$, let $\mu_n=\delta_{1/n}$ and $\mu=\delta_0$. Continuity gives $f(1/n)\to f(0)$, so $\mu_n$ converges weakly to $\mu$. However, for $A=\{0\}$,
$$
|\mu_n(A)-\mu(A)|=1
$$
for every $n$, so there is no convergence in <total variation distance>.
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