Convergent Sequence has Unique Limit

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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.


P-adic Numbers

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.