Show that $Z_p$ is sequentially compact

Question

Show that $Z_p$ is sequentially compact where $Z_p=\{a_0+a_1p+a_2p^2+...|a_i\in \{0,1,...,p-1\}\}$ and p is prime.

I want to show that if $(x_1,x_2,...)$ is a sequence in $Z_p$, then it has a cluster point.

Answer 1

Let $\alpha:\Bbb N\to\Bbb Z_p$ be a sequence.

Let $S_0=\Bbb N$. Since $\Bbb Z_p/p^i\Bbb Z_p\cong\Bbb Z/p^i\Bbb Z$ is finite, for each $i>0$ there exists $S_i\subset S_{i-1}$ such that $S_i$ is infinite and there exists $\Bbb N\ni a_i<p^i$ such that $\alpha(n)\equiv a_i\pmod{p^i}$ for each $n\in S_i$.

Put $s:n\in\Bbb N\mapsto\min S_n:\Bbb N\to\Bbb N$, so that $s$ is an increasing sequence and $\beta=\alpha\circ s$ is a subsequence of $\alpha$.

Then $\beta$ is $p$-Cauchy for if $i,j\geq N\in\Bbb N$, then $s(i)\in S_i\subseteq S_N$ and $s(j)\in S_j\subseteq S_N$ hence $\beta(i)=\alpha\circ s(i)\equiv a_N\equiv a\circ s(j)=\beta(j)\pmod{p^N}$.

Since $\Bbb Z_p$ is a closed subset of the complete $\Bbb Q_p$, it's complete, and $\beta$ converges in $\Bbb Z_p$.