Let $\displaystyle A_1, A_2, A_3, . . .$ be countable sets, and let their Cartesian product $\displaystyle A_1$ x $\displaystyle A_2$ x $\displaystyle A_3$ x ..... be defined to be the set of all sequences $\displaystyle (a_1, a_2, . . .)$ where $\displaystyle a_k$, is an element of $\displaystyle A1$. Prove that the Cartesian product is uncountable. Show that the same conclusion holds if each of the sets $\displaystyle A1, A2, . ..$ has at least two elements.