$\tau$ is an $\mathcal{F}_n$ stopping time if and only if $\{\tau=n\}\in\mathcal{F}_n$ for all $n\in\mathbb{N}$

Question

Context: these lecture notes, exercise 1.16. Not a homework exercise.

$\{\mathcal{F}_n\}_{n\geq 0}$ is a filtration.

We are given, in definition 1.15, that a non-negative, integer-valued random variable $\tau$ is an $\mathcal{F}_n$ stopping time if $\{\tau\leq n\}\in\mathcal{F}_n$ for all $n\in\mathbb{N}$.

Exercise 1.16 asks to prove what I put here as the title of this question.

One direction is obvious, as equality is stronger than $\leq$. But what about the other direction? Say we have $\tau$ such that $\{\tau< n\}\in\mathcal{F}_n$ for all $n\in\mathbb{N}$. This fits the definition of $\tau$ being a $\mathcal{F}_n$ stopping time, yet we don't have $\{\tau=n\}\in\mathcal{F}_n$. So what's stopping this scenario from happening?

Answer 1

$\{\tau\leq n\}=\cup_{i=0}^n\{\tau=i\}$ so $\{\tau\leq n\}$ is $\mathcal{F}_n$-measurable since it is a countable union of $\mathcal{F}_n$-measurable events.