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 $A$.
Definition 1
$\sequence {x_k}$ converges to the limit $l \in A$ if and only if:
- $\forall \epsilon \in \R_{>0}: \exists N \in \R_{>0}: \forall n \in \N: n > N \implies \map d {x_n, l} < \epsilon$
Definition 2
$\sequence {x_k}$ converges to the limit $l \in A$ if and only if:
- $\forall \epsilon \in \R_{>0}: \exists N \in \R_{>0}: \forall n \in \N: n > N \implies x_n \in \map {B_\epsilon} l$
where $\map {B_\epsilon} l$ is the open $\epsilon$-ball of $l$.
Definition 3
$\sequence {x_k}$ converges to the limit $l \in A$ if and only if:
- $\ds \lim_{n \mathop \to \infty} \map d {x_n, l} = 0$
Definition 4
$\sequence {x_k}$ converges to the limit $l \in A$ if and only if:
- for every $\epsilon \in \R_{>0}$, the open $\epsilon$-ball about $l$ contains all but finitely many of the $x_k$.
Note on Domain of $N$
Some sources insist that $N \in \N$ but this is not strictly necessary and can make proofs more cumbersome.
Notation
When $\sequence {x_k}$ converges to the limit $l \in A$ we can write:
- $x_n \to l$ as $n \mathop \to \infty$
or:
- $\ds \lim_{n \mathop \to \infty} x_n \to l$
This is voiced:
- As $n$ tends to infinity, $x_n$ tends to (the limit) $l$.
Some use the notation
- $\ds \lim_{n \mathop \to \infty} x_n = l$
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.
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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
- 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.
- Results about convergent sequences in the context of metric spaces can be found here.