Proof of Expectation of a Sequence

Question

Having a little trouble going about getting the answer to this question.

Show that if $|X_n - X| \leq Y_n $ and $E(Y_n \rightarrow 0) $ as n increases, then $E(X_n) \rightarrow E(X)$.

Any hints?

I'm thinking that I split $|X_n-X|$ into two cases: $(X_n -X)$ positive and negative. For the positive case, you say that since $E(Y_n) \rightarrow 0$, $E(|X_n-X|) \rightarrow 0$ too, and you remove the absolute value signs and shuffle things around.

Thinking of doing the same thing for the negative case, either:

1. $-X_n-X$
2. $X_n-X$
3. $-X_n+X$

And taking the expectation there.

Is this a correct approach?

Answer 1

\begin{align} \left|\mathbb E[X_n] - \mathbb E[X]\right| &\leqslant \mathbb E\left[|X_n-X|\right]\leqslant \mathbb E[Y_n]\stackrel{n\to\infty}\longrightarrow0. \end{align}