Confusion over Motivating Example for a $\sigma$ Algebra

Let $A_n$ be the event that the first head occurs at toss $2n$ and let $\mathcal A$ be the algebra generated by them.

Then: $$A=\bigcup_{n=1}^{\infty}A_n$$

This is an "interesting event" for us but unfortunately $A\notin\mathcal A$.

We can repair this by working instead with the $\sigma$-algebra generated by the $A_n$.