Suppose $W$ is a one-dimensional standard Brownian motion defined on some probability space $(\Omega,\mathcal F,\mathbb P)$ and let $X(t): = \exp\{ W(t) -\frac12 t-\frac{1}{t+1}\}$ for $t \ge0$. Note that $X(\infty) :=\limsup_{t\to\infty} X(t) = 0 $ a.s. because $\lim_{t\to\infty} \frac{W(t)}{t} =0$ a.s.
My question is: How to show that $\mathbb{E}[\sup_{0\le t \le \infty} X(t)] =\infty$? Many thanks.