Proving that $ \sup E[X_n] \geq E[X]$

Question

Consider the sequence of random variables $\{X_n\}_{n\geq1}$ such that $X_n$ are non-negative and $X_n \rightarrow X$ almost surely, with $\sup E[X_n]<\infty$.

Prove $E[X]\leq \sup E[X_n].$

My attempt

$$E[X]=E[\lim X_n]=\lim E[X_n]\leq \sup E[X_n]$$

Is this correct? I would appreciate any suggestion about.

Answer 1

You did not say in what sense $X_n \to X$. Also $E(\lim X_n)=\lim EX_n$ is false in general.

Assuming that $X_n \to X$ almost surely we get $EX =E\lim \inf X_n \leq \lim \inf EX_n$ by Fatou's Lemma and $\lim \inf EX_n \leq \sup_n EX_n$.

For Fatou's Lemma see https://en.wikipedia.org/wiki/Fatou%27s_lemma