I would like to prove the following
Let $(X_t)_t$ be a continuous martingale. Prove that any sequence of increasing stopping times (with values in $\mathbb{R}_{+}$) is sufficient to get a local martingale.
Here is my attempt : consider $\tau_n$ a sequence of increasing stopping time converging to $+\infty$ almost surely. We have for any $n\in\mathbb{N}$ and $s>t$
$$ \mathbb{E}[1_{\tau_n>0}X_{s}^{\tau_n} | \mathcal{F}_t] =1_{\tau_n>0} \mathbb{E}[X_{s}^{\tau_n} | \mathcal{F}_t] = 1_{\tau_n>0}\left(\mathbb{E}[X_{s}^{\tau_n}1_{\tau_n>t} | \mathcal{F}_t] + X_{\tau_n}1_{\tau_n\leq t}\right) $$
We notice that for $t$ is a stopping time bounded by $\min(tau_n, s)\geq s$. Since $X_t$ is a martingale we can apply the stopping theorem for bounded stopping time to get
$$ 1_{\tau_n>0}\left(\mathbb{E}[X_{s}^{\tau_n}1_{\tau_n>t} | \mathcal{F}_t] + X_{\tau_n}1_{\tau_n\leq t}\right) =1_{\tau_n>0}\left( X_t 1_{\tau_n>t}+ X_{\tau_n}1_{\tau_n\leq t}\right) = X_{t}^{\tau_n} $$
I would like to know if this is correct please !
Thank you !