Answer to Question #13014 in Real Analysis for endale

Question

show that the union of finite set is finite

Answer 1

Let A and B be finite sets so that their elements could be numbered:

A = {a1, a2, ..., an},
B = {b1, b2, ..., bm}.

Let C = A U B and c1 = a1, c2 = a2, ..., cn = an, c(n+1) = b1, c(n+2) = b2, ..., c(n+m) = bm. Then

C = {c1, c2, ..., c(n+m)}.

So, C is finite. Similarly we can add another finite set to obtained set C. So, finite union of finite sets is finite.