Suppose $\displaystyle {X(t,\omega)}$ is a stochastic process on a filtered probability space $\displaystyle (\Omega,\mathbb{F},P)$, and let $\displaystyle A$ be any Borel set.

Then, I want to show that the first date for which $\displaystyle X(t,\omega)\in A$ is a stopping time.

The attached pdf file is my proof. Is it complete?