Argue that if $X$ and $Y$ are nonnegative random variables, then

$$ \E[\max(X, Y)] \le \E[X] + \E[Y] $$

There is a hidden assumption that $X$ and $Y$ are on the same domain (otherwise, you can't define $\max(X, Y)$). The expectations can be expanded as this:

$$ \E[\max(X, Y)] = \sum n \cdot \max(\Pr\{X = n\}, \Pr\{Y = n\}) \\ \E[X] = \sum n \Pr\{X = n\} \qquad \E[X] = \sum n \Pr\{Y = n\} $$

Each of the summands in the first formula appears in either $\E[X]$ or $\E[Y]$. Their sum contains twice as many, all nonnegative. That makes it equal or greater.