Equivalence of Definitions of Convergent Sequence in Metric Space/Definition 2 implies Definition 4

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

Let $l \in A$.

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

Let $\sequence {x_k}$ satisfy:
 * $\forall \epsilon > 0: \exists N \in \R_{>0}: \forall n \in \N: n > N \implies x_n \in \map {B_\epsilon} l$

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

Then:
 * for every $\epsilon \in \R{>0}$, the open $\epsilon$-ball about $l$ contains all but finitely many of the $p_n$.

Proof
Let a fixed $\epsilon \in \R{>0}$ be selected.

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

Hence the only $x_k$ that cannot be in the open $\epsilon$-ball $\map {B_\epsilon} l$ of $l$ are those for which $n \le N$.

There are finitely many of these.