Equivalence of Definitions of Convergent Sequence in Metric Space/Definition 1 iff Definition 2

Theorem

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

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

The following definitions of the concept of Convergent Sequence in the context of Metric Spaces are equivalent:

$\sequence {x_k}$ converges to the limit $l \in A$ if and only if:

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

$\sequence {x_k}$ converges to the limit $l \in A$ if and only if:

$\forall \epsilon > 0: \exists N \in \R_{>0}: \forall n \in \N: n > N \implies x_n \in \map {B_\epsilon} l$

where $\map {B_\epsilon} l$ is the open $\epsilon$-ball of $l$.

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.

$\blacksquare$