Topology/Sequences

A sequence in a space $X$ is defined as a function from the set of natural numbers into that space, that is $f:\mathbb{N}\to X$. The members of the domain of the sequence are $f(1),f(2),\dots$ and are denoted by $f(n)=a_n$. The sequence itself, or more specifically its domain are often denoted by $⟨a_i⟩$.

The idea is that you have an infinite list of elements from the space; the first element of the sequence is $f(1)$, the next is $f(2)$, etc. For example, consider the sequence in $ℝ$ given by $f(n)=1/n$. This is simply the points $1,1/2,1/3,1/4,...$ Also, consider the constant sequence $f(n)=1$. You can think of this as the number 1, repeated over and over.

Convergence

Let $X$ be a set and let $𝒯$ be a topology on $X$
Let $⟨x_i⟩$ be a sequence in $X$ and let $x\in X$

We say that "$⟨x_i⟩$ converges to $x$" if for any neighborhood $U$ of $x$, there exists $N\in \mathbb{N}$ such that $n\in \mathbb{N}$ and $n>N$ together imply $x_n\in U$

This is written as $\underset{n\to \mathrm{\infty }}{\lim }{x}_n=x$

Exercises

1. Give a rigorous description of the following sequences of natural numbers:
   (i) $1,2,3,4,5\dots$
   (ii) $2,-4,6,-8,10,\dots$
   
2. Let $X$ be a set and let $𝒯$ be a topology over $X$. Let $x\in X$ and let $U$ be a neighbourhood of $x$.
   Let $U_1\subset U$ and $x\in U_1$. Similarly construct neighbourhoods $U_i\subset U_{i-1}$ with $x\in U_i\mathrm{\forall }i$. Let $⟨x_i⟩$ be a sequence such that each $x_i\in U_i$.
   
   Prove that $\underset{n\to \mathrm{\infty }}{\lim }{x}_n=x$

Template:Navigation Template:Subjects