Definition:Convergent Sequence/Metric Space

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Definition

Let $M = \left({A, d}\right)$ 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:

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

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:

$\displaystyle \lim_{n \mathop \to \infty} \map d {x_n, l} = 0$


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 $p_n$.


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 $N$.


Note on Domain of $N$

Some sources insist that $N \in \N$ but this is not strictly necessary and can make proofs more cumbersome.


Also see