Proving Bound on Expected Value of Sup of Sequence of R.V.

Question

I am trying to prove the following:

Let ${X_n}$ be a sequence of non-negative independent random variables. Show that:

$E(\sup X_{n}) \le \sum_{n=1}^{\infty} E(X_{n}^{2})+1$,

where the sup is taken over the index n.

Not sure how to tackle this problem.

Answer 1

If $b_1,b_2,\ldots$ are non-negative reals, then $\sup_nb_n\le \sum_{n=1}^\infty b_n^2 +1$.