Convergence in Total Variation Implies Convergence in Distribution Suppose $X,Y$ are random variables. We define the total variation distance of random variables to be
$d(X,Y)= \inf \{P(|X′−Y′|>0): X′,Y′$ are couplings of $ X,Y$ respectively$\}$.
Does convergence in total variation imply convergence in distribution? 
My feeling is that it should but I'm having trouble proving it.
Thanks!

Let us use the dual representation of the total variation distance:
$$
d_{TV}(X,Y) = \sup_{f: \mathcal X \to [-1,1]} Ef(X) - Ef(Y)
$$
Then for every positive (measurable) bounded $f$:
\begin{align}
Ef(X_n)&\le Ef(X) +  \sup_{\mathcal X} |f|\times d_{TV}(X_n,X)\\
Ef(X_n)&\ge Ef(X) -  \sup_{\mathcal X} |f|\times d_{TV}(X_n,X)
\end{align}
