Convergent Sequence in P-adic Numbers has Unique Limit

Theorem
Let $p$ be a prime number.

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

Let $\sequence {x_n} $ be a sequence in $\Q_p$.

Then $\sequence {x_n}$ can have at most one limit.

Proof
From P-adic Metric on P-adic Numbers is Non-Archimedean Metric the $p$-adic metric is a metric on $\Q_p$.

By definition, the sequence $\sequence {x_n}$ converges in $\Q_p$ :
 * $\sequence {x_n}$ converges in the $p$-adic metric.

The result then follows from Convergent Sequence in Metric Space has Unique Limit.