I am trying to prove the following:
Let $(X_n)$ be a martingale with respect to $(\mathcal{F}_n)$ and suppose $\tau_1$ and $\tau_2$ are bounded stopping times such that $\tau_1\le \tau_2 < B$, where $B < \infty$ is the bound. Then $E(X_{\tau_2}|\mathcal{F}_{\tau_1}) = X_{\tau_1}.$
What I've learned so far are definition and some properties of conditional expectation and martingales and I think no advanced knowledge is necessary to solve this.
But I could not combine that definition and properties to produce good solution to this.
If anyone has any tips on how to do, I'd appreciate it a lot.