Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space.
If $X$ and $Y$ are integrable random variables, then $X + Y$ is an integrable random variable.
Here is my attempt.
To begin with, let us notice that $X = X^{+} - X^{-}$ and $Y = Y^{+} - Y^{-}$. Since $X$ and $Y$ are integrable, it means that $\mathbb{E}(|X|) < +\infty$ and $\mathbb{E}(|Y|) < +\infty$. We have to prove that $E(X + Y) < +\infty$. Indeed, given that $X^{+} \geq 0$ and $X^{-}\geq 0$, there are two sequences of non-negative and non-decreasing random variables $X^{+}_{n}$ and $X^{-}_{n}$ such that $X^{+}_{n}\to X^{+}$ and $X^{-}_{n}\to X^{-}$. Similarly, we can also state there exist two sequences of non-negative and non-decreasing random variables $Y^{+}$ and $Y^{-}$ such that $Y^{+}_{n}\to Y_{n}$ and $Y^{-}_{n}\to Y^{-}$. Hence, the monotone convergence theorem ensures us that $\mathbb{E}(X^{+}_{n})\to\mathbb{E}(X^{+})$ and $\mathbb{E}(X^{-}_{n})\to\mathbb{E}(X^{-})$ (similar results for $Y^{+}$ and $Y^{-}$).
Finally, gathering these results and the assumption given in the problem setting, one concludes that \begin{align*} \mathbb{E}(|X + Y|) & = \mathbb{E}(|X^{+} - X^{-} + Y^{+} - Y^{-}|)\\\\ & \leq \mathbb{E}(X^{+} + X^{-} + Y^{+} + Y^{-})\\\\ & = \mathbb{E}(\lim_{n\to\infty}X^{+}_{n} + \lim_{n\to\infty}X^{-}_{n} + \lim_{n\to\infty}Y^{+}_{n} + \lim_{n\to\infty}Y^{-}_{n})\\\\ & = \mathbb{E}(\lim_{n\to\infty} X^{+}_{n}) + \mathbb{E}(\lim_{n\to\infty} X^{-}_{n}) + \mathbb{E}(\lim_{n\to\infty} Y^{+}_{n}) + \mathbb{E}(\lim_{n\to\infty} Y^{-}_{n})\\\\ & = \mathbb{E}(X^{+}) + \mathbb{E}(X^{-}) + \mathbb{E}(Y^{+}) + \mathbb{E}(Y^{-})\\\\ & = \mathbb{E}(|X|) + \mathbb{E}(|Y|) < +\infty \end{align*}
Can someone please check out my solution?