Convergent Sequence in Normed Vector Space has Unique Limit

Theorem
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.

Proof
$\ds \lim_{n \mathop \to \infty} x_n = L_1$ and $\ds \lim_{n \mathop \to \infty} x_n = L_2$ such that $L_1 \ne L_2$.

Let $\epsilon = \dfrac {\norm {L_1 - L_2} } 3$.

From properties of norm it follows that $\epsilon > 0$.

By definition:


 * $\exists N_1 \in \N : \forall n > N_1 : \norm {x_n - L_1} < \epsilon$


 * $\exists N_2 \in \N : \forall n > N_2 : \norm {x_n - L_2} < \epsilon$

Choose $n > N_1 + N_2$.

Then:

which implies that:


 * $1 < \dfrac 2 3$

This is a contradiction.

Hence $L_1 = L_2$.

Also see

 * Convergent Sequence has Unique Limit