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 $\lim_{n\to\infty}x_n=x$