Sequence of P-adic Integers has Convergent Subsequence/Proof 2

Proof
From P-adic Integers are Compact Subspace:
 * $\Z_p$ is a compact subspace in the metric space induced by $\norm{\,\cdot\,}_p$

From Compact Subspace of Metric Space is Sequentially Compact in Itself:
 * $\Z_p$ is sequentially compact in itself

By definition of sequentially compact in itself:
 * every sequence in $\Z_p$ has a subsequence which converges in the topology to a point in $\Z_p$

From Equivalence of Definitions of Convergence in Normed Division Rings:
 * every sequence in $\Z_p$ has a subsequence which converges in the norm to a point in $\Z_p$