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.
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.