Definition:Convergent Sequence/Metric Space

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

Definition 3
We can write:
 * $x_n \to l$ as $n \to \infty$

or:
 * $\displaystyle \lim_{n \to \infty} x_n \to l$

This is voiced:
 * As $n$ tends to infinity, $x_n$ tends to (the limit) $l$.

If $M$ is a metric space, some use the notation
 * $\displaystyle \lim_{n \to \infty} x_n = l$

This is voiced:
 * The limit as $n$ tends to infinity of $x_n$ is $l$.

Note, however, that one must take care to use this alternative notation only in contexts in which the sequence is known to have a limit.

It follows from Sequence Converges to Point Relative to Metric iff it Converges Relative to Induced Topology that this definition is equivalent to that for convergence in a topological space.

Comment
The sequence $x_1, x_2, x_3, \ldots, x_n, \ldots$ can be thought of as a set of approximations to $l$, in which the higher the $n$ the better the approximation.

The distance $d \left({x_n, l}\right)$ between $x_n$ and $l$ can then be thought of as the error arising from approximating $l$ by $x_n$.

Note the way the definition is constructed.


 * Given any value of $\epsilon$, however small, we can always find a value of $N$ such that ...

If you pick a smaller value of $\epsilon$, then (in general) you would have to pick a larger value of $N$ - but the implication is that, if the sequence is convergent, you will always be able to do this.

Note also that $N$ depends on $\epsilon$. That is, for each value of $\epsilon$ we (probably) need to use a different value of $N$.