Topology/Sequences

A sequence in a space ${\displaystyle X}$ is defined as a function from the set of natural numbers into that space, that is ${\displaystyle f:\mathbb {N} \to X}$. The members of the domain of the sequence are ${\displaystyle f(1),f(2),\ldots }$ and are denoted by ${\displaystyle f(n)=a_{n}}$. The sequence itself, or more specifically its domain are often denoted by ${\displaystyle \left\langle a_{i}\right\rangle }$.

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

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

We say that "${\displaystyle \left\langle x_{i}\right\rangle }$ converges to ${\displaystyle x}$" if for any neighborhood ${\displaystyle U}$ of ${\displaystyle x}$, there exists ${\displaystyle N\in \mathbb {N} }$ such that ${\displaystyle n\in \mathbb {N} }$ and ${\displaystyle n>N}$ together imply ${\displaystyle x_{n}\in U}$

This is written as ${\displaystyle \lim _{n\to \infty }x_{n}=x}$

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