Let $\tau$ and $\rho$ be stopping times with respect to filtration $(\mathbb{F}_t)_{t\ge0}$ and $A \in \mathbb{F}_\tau$. Show that $$A \cap\{\tau\le\rho\} \in \mathbb{F}_\rho$$
My work so far
Firstly let's consider discrete case, then :
$$\{\tau \le \rho\} = \bigcup_{s \le t}\{\tau=s\}\cap\{\rho\le t-s\}$$
First term of sum is in $\mathbb{F}_\tau$ and second one in $\mathbb{F_{\tau-p}} \subset \mathbb{F}_\tau$.
So the whole sum is in $\mathbb{F}_\tau$ so that $\{\tau \le \rho\}$. Also if $A \in \mathbb{F}_\tau$ then $A \cap\{\tau \le \rho\} \in \mathbb{F}_\tau$. where is my mistake ?