A premise. Let $X_n$ be a sequence of random variables and let $X$ be a random variable. Call
$$
F_n(x)=\mathbb{P}[X_n\leq x]
$$
and
$$
F(x)=\mathbb{P}[X\leq x].
$$
I say that $X_n$ converges in distribution to $X$ and I write
$$
X_n\stackrel{d}{\to} X
$$
if and only if
$$
F_{n}(x)\to F(x)
$$
for all the $x$ in which $ F(x)$ is continuous.
My problem. I am looking for a simple proof of this result (which is part of the more general Portmanteau theorem).
Theorem. A sequence of random variables $X_n$ converges in distribution to $X$ if and only if $$ \lim_{n\rightarrow\infty}\mathbb{E}\left[f\left(X_n\right)\right]=E[f(X)] $$ for any bounded continuous function $f$.
Any suggestion of textbooks/lectures?
Here is a sketch.
You need to show that for each $\epsilon>0$, the inequality $|Ef(X_n)-Ef(X)|<\epsilon$ holds for all $n$ sufficiently large. You can start out by picking $K$ so that $P(|X|>K)<\epsilon/(2\|f\|).$ There is a finite set of intervals covering the compact set $[-K,K]$ such that $f$ is within $\epsilon/2$ of a constant on each of them. You can make all the above choices so that all the endpoints are points of continuity of $F$. Now replace $f$ with a piecewise constant function suggested by the above partition of $\mathbb R$ into intervals, upper bounding $|Ef(X_n)-Ef(X)|$ by a quantity that converges to something less than $\epsilon$.