Convergent Sequence in P-adic Numbers has Unique Limit
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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 $\Q_p$ if and only if:
- $\sequence {x_n}$ converges in the $p$-adic metric.
The result then follows from Convergent Sequence in Metric Space has Unique Limit.
$\blacksquare$