Let $(\Omega, \mathcal{A},(\mathcal{F}_n)_{n≥1},\mathbb{P})\;$ be a given filtrated probability space, and $X = (X_n)_{n≥1}$ , $Y = (Y_n)_{n≥1}\;$ be two $(\mathcal{F}_n)_{n≥1}$-martingales. Set $$ \tau := \min \left\{ n \geq 1 : X_n = Y_n \right\} $$ How we can show that $\tau\;$ is an $(\mathcal{F}_n)_{n≥1}$-stopping time ?
And let $$ Z_n:=(X_n-Y_n)\mathbf{1}_{\left\{ \tau \geq n \right\}}$$ How can we show that $ (Z_n)_{n≥1}$ is an $(\mathcal{F}_n)_{n≥1}$-martingale ?