How to prove that "The union of a countably infinite collection of (pairwise different) nonempty sets is infinite"?

Question

Consider the following proposition

The union of a countably infinite collection of (pairwise different) nonempty sets is infinite.

I think it is correct. But how to prove it rigorously?

Answer 1

As requested in the comments:

Suppose the claim were false. That is, suppose that the union was a finite set $S$. Then every set in your collection would be a subset of the finite set $S$ but there are only $2^{|S|}-1$ of these (you have excluded the empty set). Thus we have a contradiction.