Let $E=\{u\subset \mathbb{R} : \text{$u$ is finite or $\complement u$ is finite}\}$ Prove that $E$ is a sigma algebra on $\mathbb{R}$.
I had checked the three conditions of a sigma algebra, and actually I got the proof.
But I'm confused because for example $\{A_i=\{i\}\}$ belong to $E$ for all $i$, but the union doesn't! And that doesn't match the condition of closed under countable union. What is my mistake? I think my example is false, but why?
As pointed out by José Carlos Santos, you proved that $E$ is not a $\sigma$-algebra.
A small modification of the definition of $E$ would make is a $\sigma$-algebra, namely, replacing the word "finite" by "at most countable".