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

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Theorem

Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers for some prime $p$.

Let $\sequence{x_n}$ be a sequence of $p$-adic integers.

Then:

there exists a convergent subsequence $\sequence {x_{n_r} }_{r \mathop \in \N}$ of $\sequence{x_n}$

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 $\Z_p$

$\blacksquare$

Sources

• 1997: Fernando Q. Gouvea: p-adic Numbers: An Introduction ... (previous) ... (next): $\S 3.3$ Exploring $\Q_p$: Problem $107$