An inequality for the mgf using Jensen’s inequality Given non-negative random variables $X_1,X_2,...$ how to show that 
$$\mathbb{E}\exp(t\max\limits_{1\leq i\leq n}X_i)\leq \sum\limits_{1\leq i\leq n}\mathbb{E}\exp(tX_i).$$
I think we should start with 
$$\max\limits_{1\leq i\leq n}X_i\leq \sum\limits_{1\leq i\leq n}X_i$$
and the apply Jensen's inequality, but I need help with clarification of the details

If $t<0$ then $t \max_k X_k(\omega) \le t X_i(\omega)$ for all $i$ and if
$t \ge 0$ then $ t \max X_k(\omega) \le tX_i(\omega)$ for some $i$.
Hence $e^{t \max_k X_k(\omega) } \le \sum_k e^{t X_k(\omega)}$ and hence taking expectations
we have the desired result.
