Snell envelope and optimal stopping time Suppose $(G_n)_{0\leq n\leq N}$ is a process adapted to a filteing $(\mathcal{F}_n)_{0\leq n\leq N}$.

The Snell envelope of $(G_n)$ is the smallest supermatingale dominates $(G_n)$. It's defined as follows:$$S_N=G_N,S_{n}=\max\{G_n,E[S_{n+1}|\mathcal{F}_n]\},n=N-1,\ldots,0.$$

The stopping time $\tau=\inf_k\{0\leq k\leq N:S_k=G_k\}$ is the optimal stopping time to maximize the gain $EX_\tau$.

It can also be shown that $S_{n\wedge\tau}$ is a martingale, my question is, what is the significance of this result? We have already find the optimal rule( $\tau$), and we can compute the optimal gain $EX_\tau=ES_0$, so what can we say about the conclusion that $S_{n\wedge \tau}$ is a martingale?
I'll assume that you're using $X$ and $G$ interchangeably.

The reason is that, to prove that for the stopping time $\tau^*$ is optimal (such that $E [G_{\tau^*}] = sup_{\tau \in [0, T]} E [G_\tau]$), both conditions need to be satisfied:


$S_{\tau^*} = G_{\tau^*}$ a.s., and
that the stopped sequence $S_{n \wedge \tau^*}$ is a martingale.


Since $S_n$ dominates $G_n$, the first condition only tells us that the process $G_n$ for $n \geq \tau^*$ is a supermartingale. 

This but tells us nothing about the process $G_n$ for $n \lt \tau^*$, except that we know that it is dominated by $S_n$. Since $E[S_{n}] \geq E[S_{\tau^*}]$ for $n \lt \tau^*$, it is entirely possible that $E[G_{n} \gt E[G_{\tau^*}]$ for some $n \lt \tau^* $, which would mean that ${\tau^*}$ is not optimal. 

But, if the stopped process $S_{n \wedge \tau^*}$ for $\forall n \in [0, T]$ is a martingale, then, for $n \lt \tau^*$, $E[S_{n}] = E[S_{\tau^*}]$ and, by the fact that $G_n$ is dominated by $S_n$, $E[G_{n}] \leq E[G_{\tau^*}]$. This confirms that $\tau^*$ is optimal.

Please note that this only proves that $\tau^*$ is an optimal stopping time, not that it is the smallest optimal stopping time.
This can be used when pricing an American option. The Snell envelope property guarantees the buyer of the option that this is the best possible price. But nobody would by a supermartingale. The second property gives the buyer the guarantee that the price is fair if he exercises optimally.