Proof of Helly Bray theorem: $X_n\overset{d}{\rightarrow}X\Leftrightarrow Eg(X_n)\rightarrow Eg(X)$ for bounded continuous $g$

Question

Helly Bray Theorem
$X_n\overset{d}{\rightarrow}X\Leftrightarrow Eg(X_n)\rightarrow Eg(X)$ for all bounded, continuous functions, $g$.

For the only if part of the proof, I am a little stuck. I have noted the following conclusions successfully but I don't know how to prove the required. It is enough to show that for some large enougn $n$, $|Eg(X_n)-Eg(X)|<\epsilon$ for any arbitrarily fixed $\epsilon>0$.

$b<c$ such that $F(b)<\epsilon,1-F(c)<\epsilon$

A sequence of numbers $b=a_0<a_1<...<a_{m-1}<a_m=c$ such that

• $a_i$ is a continuity point of $F$ for all $i$
• $|a_{i+1}-a_i|<\delta$ for all $i$ $\Rightarrow g(x)-g(a_i)<\epsilon$ for any $x\in[a_{i-1},a_i]$

$\exists M$ such that $|g(x)|<M$ for all $x$

I need to prove

1. For $h=\begin{cases}g(a_i)\quad\text{if, }a_{i-1}<x\le a_i\text{ for some }i\\0\quad\quad~~\text{otherwise}\end{cases}$, $|Eh(X_n)-Eh(X)|<\epsilon$

2. Finally, I have $|Eg(X_n)-Eg(X)|\le |Eg(X_n)-Eh(X_n)|+|Eh(X_n)-Eh(X)|+|Eh(X)-Eg(X)|$. How to conclude this is $<\epsilon$ for some large enougn $n$?