Show that if \(B\) is not finite and \(B \subset A\), then \(A\) is not finite.
Solution
Suppose, toward a contradiction, that \(A\) were finite. Every subset of a finite set is finite: if \(|A|=n\), identify \(A\) with \(\{1,\dots,n\}\); then the image of \(B\) is a subset of this finite set and hence is finite, so \(B\) is finite as well. This contradicts the hypothesis that \(B\) is not finite.
Therefore \[ B\subset A\text{ and }B\text{ infinite}\quad\Longrightarrow\quad A\text{ infinite}. \] Equivalently, this is the contrapositive of the statement that every subset of a finite set is finite.