Supremum and infimum of a stochastic process

Question

Suppose $\{X_t\}_{t\geq 0}$ is a real-valued stochastic process satisfying $E[X^2_t]\rightarrow\infty$ as $t\rightarrow\infty$. Does it hold that $\sup |X_t|\rightarrow \infty$? Intuitively I think since the second moment of $X_t$ explodes, $X_t$ must diverge either from above or from below, or both. However, I simply don't know where to start to prove it.