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 $f(1),f(2),\ldots$ and are denoted by $f(n)=a_{n}$. The sequence itself, or more specifically its domain are often denoted by $\left\langle a_{i}\right\rangle$.

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 $\mathbb {R}$ 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 ${\mathcal {T}}$ be a topology on $X$
Let $\left\langle x_{i}\right\rangle$ be a sequence in $X$ and let $x\in X$

We say that "$\left\langle x_{i}\right\rangle$ 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 $\lim _{n\to \infty }x_{n}=x$

Exercises

1. Give a rigorous description of the following sequences of natural numbers:
   (i) $1,2,3,4,5\dots$
   (ii) $2,-4,6,-8,10,\ldots$
   
2. Let $X$ be a set and let ${\mathcal {T}}$ 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 $x\in U_{i}\forall i$. Let $\left\langle x_{i}\right\rangle$ be a sequence such that each $x_{i}\in U_{i}$.
   
   Prove that $\lim _{n\to \infty }x_{n}=x$