Convergence of sequence of infinite expectations

Question

If $X_n$ is a sequence of non negative random variables decreasing to $X$ (a non negative rv ) in the limit as $n \rightarrow \infty$ and $E[X_n] = \infty $ for all $n$ then is it true that that $E[X] = \infty $ ?

Edit : If we add a condition that $|X_n - X| \leq c_n $ where $c_n$ is a sequence of non negative reals converging to 0 , then does the fact hold ?

Answer 1

The question in the edited part is obvious. $EX \geq EX_n-c_n=\infty$.

Answer for the first part: let $X$ be any non-negative random variable with $EX=\infty$. Consider the sequence $\frac X n$. This decreases to $0$ which has finite expectation.

Answer 2

This is not true. Let $Y$ be any non negative r.v. with $\operatorname{E}Y = \infty$, $X_n = Y \cdot \mathbb{I}_{Y > n}$. Then $\operatorname{E} X_n = \infty$, but $X_n \to 0$.