Education: Convergence

First, Lets translate the calculus definition of convergence, to the "language" of metric spaces: We say that a sequence

x_n

converges to

x

if for every

\epsilon > 0

exists

N

that for each

n^* > N

the following holds:

d(x_{n^*},x) < \epsilon

.
Equivalently, we can define converges using Open-balls: A sequence

x_n

converges to

x

If for every

\epsilon > 0

exists

N

that for each

n^* > N

the following holds:

x_{n^*} \in B_\epsilon(x)

The latter definition uses the "language" of open-balls, But we can do better - We can remove the

\epsilon

from the definition of convergence, thus making the definition more topological. Let's define that

x_n

converges to

x

(and mark

x_n \rightarrow x

) , if for every ball

B

around

x

, exists

N_B

that for each

n^* > N_B

the following holds:

x_{n^*} \in B(x)

x

is called the limit of the sequence.

The definitions are all the same, but the latter uses topological terms, and can be easily converted to a topological definition later.

If a sequence has a limit, it has only one limit.
Proof Let a sequence $x_n$ have two limits, $x\,$ and $x^\prime$ . If they are not the same, we must have $0<d(x,x^\prime)$ . Let $\epsilon$ be smaller than this distance. Now for some $N$ , for all $n>N$ , it must be the case that both $x_n \in B_{\epsilon / 2}(x)$ and $x_n \in B_{\epsilon / 2}(x^\prime)$ by virtue of the fact $x\,$ and $x^\prime$ are limits. But this is impossible; the two balls are separate. Therefore the limits are coincident, that is, the sequence has only one limit.
If $x_n \rightarrow x$ , then almost by definition we get that $d(x_n, x) \rightarrow 0$ . ( $d(x_n, x)$ Is the sequence of distances).

In $\mathbb{R}$ with the natural metric, The series $x_n = \frac{1}{n}$ converges to $0$ . And we note it as follows: $\frac{1}{n}\rightarrow 0$
Any space, with the discrete metric. A series $x_n$ converges, only if it is eventually constant. In other words: $x_n\rightarrow x$ If and only if, We can find $N$ that for each $n^* > N$ , $x_{n^*} = x$
An example you might already know: