# Convergent Sequence has Unique Limit

## Theorem

### Hausdorff Space

Let $T = \struct {S, \tau}$ be a Hausdorff space.

Let $\sequence {x_n}$ be a convergent sequence in $T$.

Then $\sequence {x_n}$ has exactly one limit.

### Metric Space

Let $M = \struct {A, d}$ be a metric space.

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

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

### Normed Vector Space

Let $\struct {X, \norm {\,\cdot\,} }$ be a normed vector space.

Let $\sequence {x_n}$ be a sequence in $\struct {X, \norm {\,\cdot\,} }$.

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

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.

### Real Numbers

Let $\sequence {s_n}$ be a real sequence.

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