Product of Finite Sets is Finite/Proof 1

Proof
By the definition of Cartesian product:


 * $S \times T = \set {\tuple {s, t}: s \in S, t \in T}$

Then by definition of set union:


 * $S \times T = \displaystyle \bigcup_{s \mathop \in S} \set s \times T$

Also, the mapping $g_s: \set s \times T \to T$ defined by:


 * $\map {g_s} {s, t} = t$

is a bijection.

Therefore, since $T$ is finite, so is $\set s \times T$ for all $s \in S$.

Since $S$ is finite, the result follows from Finite Union of Finite Sets is Finite.