Equivalence of Definitions of Convergent Sequence in Metric Space/Definition 1 iff Definition 3

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

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

Proof
By definition of a convergent real sequence:
 * $\ds \lim_{n \mathop \to \infty} \map d {x_n, l} = 0$


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

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

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.