In a local martingale, does the sequence of stopping times is important? Let $(M_t)$ a local martingale. Does : 
1) There exist an increasing sequence of stopping time $(\tau_k)_k$ s.t. $\tau_k\to \infty $ a.s when $k\to \infty $ s.t. the process $(M_{\tau_k\wedge t})_t$ is a martingale 
or 
2) For any increasing sequence of stopping time $(\tau_k)_k$ s.t. $\tau_k\to \infty $ a.s. when $k\to \infty $, the process $(M_{\tau_k\wedge t})_t$ is a martingale ?

Consider the local martingale $X_t$ constructed in these notes, which consists of standard Brownian motion stopped upon hitting 1 and sped up to fit all of time into $(0,1)$.  If you take $\tau_k = k$ for $k = 1, 2, 3, \dots$, then $\tau_k \uparrow \infty$, but $X_{\tau_k \wedge t}$ is the same as $X_t$ for all $k$, and is not a martingale.
