21/10/2014

Convergence

                   Convergence:                          




Definition

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 > 0exists 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_nconverges 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.

Properties

  • 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).

Examples

  • 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^* > Nx_{n^*} = x
  • An example you might already know:
The space \mathbb{R}^k For any p-norm induced metric, when p\geq 1. Let \vec{x_n} =  (x_{n,1},x_{n,2},\cdots, x_{n,k}). and let \vec{x} =  (x_{1},x_{2},\cdots, x_{k}).
Then, \vec{x_n} \rightarrow \vec{x} If and only if  \forall i, 1\leq i \leq k: x_{n,i} \rightarrow x_{i}.

Uniform Convergence

A sequence of functions \{ f_n \} is said to be uniformly convergent on a set S if for any \epsilon>0, there exists an N such that when a and b are both greater than N, then d(f_a(x),f_b(x)) < \epsilon for any x \in S.