Random walk - first visit in zero is a random variable proof

Question

Define $\tau = inf\{ n \geq 1: S_n = 0\}$. Prove that $\tau$ is a random variable.

This has probably been already proved many times. I am looking for a reference proof. Thank you.

Answer 1

$\tau$ only takes values in $\{1,2,3,\dots,+\infty\}=\mathbb N_+\cup\{+\infty\}$ so it is enough to prove that $\{\tau=n\} $ is measurable for every $n\in\mathbb N_+$.

This follows directly from: $$\{ \tau=n\} =\{ S_{1}\neq0\} \cap\cdots\cap\{ S_{n-1}\neq0\} \cap\{ S_{n}=0\} $$