Definition:Convergent Sequence/Metric Space

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

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

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 $\map d {x_n, l}$ 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 $N$.

Also see

 * Equivalence of Definitions of Convergent Sequence in Metric Space


 * Sequence Converges to Point Relative to Metric iff it Converges Relative to Induced Topology for a proof that this definition is equivalent to that for convergence in the induced topology.