Question about algebra and σ–algebra

Question

Let A be an algebra of subsets of a set X. If A is finite, prove that A is in fact a σ–algebra. How many elements can A have?

Answer 1

Consider checking that the union of a sequence $B_1,B_2,\ldots$ of elements of $A$ is also an element of $A$. Because $A$ is finite, there are many repeats in that sequence!
More precisely, if we define $C=\{B\in A: B=B_n$ for some $n\ge 1\}$, then $C$ is a finite subset of $A$, and $\cup_{n=1}^\infty B_n =\cup_{B\in C} B$. This latter is the union of finitely many elements of $A$, so it is an element of $A$.