Proving almost sure finiteness of a stopping time

Question

Let $M_t$ be a local martingale and $S_t = \sup_{0 \leq s \leq t}M_s$ its running supremum. How can I show that the stopping time $T=\inf\{t \geq 0 : S_t - M_t = a\}$ for an arbitrary $a>0$ is almost surely finite given the fact that the quadratic variation of M almost surely diverges to infinity $(<M>_{\infty}=\infty)$ ?