# Proving the Monotonic Convergence Theorem without Assuming Non-Negative RVs

I'm faced with the following problem as part of a larger homework problem and do not really know how to go about it:

• let $X_n$ be an increasing sequence of random variables (that could be negative).
• assume that $\lim_{n\rightarrow\infty}X_n = X$ a.e.
• assume there exists a random variable $Y$ such that $X_n \geq Y$ for all $n \in \mathbb{N}$ a.e.
• assume that $\mathbb{E}[X_n] < \infty$ for all $n$, $\mathbb{E}[X] < \infty$, and $\mathbb{E}[Y] < \infty$

With these assumptions in place, I'm trying to show that $\lim_{n\rightarrow\infty}\mathbb{E}[X_n] = \mathbb{E}[X]$, but I can't do it because everything I know about random variables requires that either the $X_n$ or the $Y$ are non-negative.

Anyone with an idea of how to proceed?

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You may want to consider $\tilde{X}_n = X_n - Y$ and $\tilde{X} = X - Y$. – Sangchul Lee Oct 24 '12 at 2:25
@sos440 Thanks for this! For some reason I thought it would much more complicated. The basic idea is that the Monotonic Convergence Theorem holds for $\tilde{X}_n$ they are non-negative. Therefore, we can use it to obtain $E[\tilde{X}_n] \uparrow E[\tilde{X}]$. The desired result then follows as a consequence of the finite expectations of $X,X_n$ and $Y$- right? – Elements Oct 24 '12 at 3:05
Yes, exactly. The finiteness of $Y$ plays a key role in this proof. – Sangchul Lee Oct 24 '12 at 3:07