Equivalence of Definitions of Convergent Sequence in Metric Space

Theorem
Let $M = \left({A, d}\right)$ be a metric space or a pseudometric space.

Let $\sequence {x_k}$ be a sequence in $A$.

Definition 1 iff Definition 2
By definition of an open ball:
 * $\forall n \in \N: \map d {x_n, l} < \epsilon \iff x_n \in \map {B_\epsilon} l$

The result follows.

Definition 1 iff Definition 3
By definition of a convergent real sequence:
 * $\displaystyle \lim_{n \mathop \to \infty} \map d {x_n, l} = 0$


 * $\forall \epsilon \in \R_{>0}: \exists N \in \R_{>0}: n > N \implies \size {\map d {x_n, l} - 0} < \epsilon$

From Distance in Pseudometric is Non-Negative, for all $x, y \in A: \map d {x, y} \ge 0$.

Hence:
 * $\forall n \in \N: \map d {x_n, l} = \size {\map d {x_n, l}} = \size {\map d {x_n, l} - 0}$

The result follows.