Prove that if A1, A2, ...,An are finite sets, then the union from i=1 to n of Ai is finite.?

Answer 1

Assume that the numbers of elements of A1, A2, ..., An are m1, m2, ..., mn, respectively. Then their union has at most m1+m2+...+mn elements (exactly so many if the sets are disjoint, but if they have common elements then their union has less then m1+m2+...+mn elements), which is finite, because it is the sum of finitely many finite numbers.

Answer 2

Sum of the finite is still finite.