$\mathbb{Z}_p=\varprojlim \mathbb{Z}/p^{n}\mathbb{Z}$ is uncountable.

Question

The ring of $p$-adic integers is given by $\mathbb{Z}_p=\varprojlim \mathbb{Z}/p^{n}\mathbb{Z}$. From this description how can we conclude that $\mathbb{Z}_p$ is uncountable ?

It follows from the description that each nonzero element of $\mathbb{Z}_p$ is of infinite order (as an element of component wise additive group $\mathbb{Z}_p$). But I cannot produce any contradiction assuming countability of $\mathbb{Z}_p$. Help me.

Answer 1

Each element of $\Bbb Z_p$ has a "base-$p$" expansion $$a_0+pa_1+p^2a_2+\cdots$$ with each $a_j\in\{0,1,\ldots,p-1)$. Now use the Cantor diagonalisation argument.