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
(i) $$1,2,3,4,5\dots$$ (ii) $$2,-4,6,-8,10,\ldots$$
 * 1) Give a rigorous description of the following sequences of natural numbers:

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 $$x_i\in U_i$$.
 * 1) 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 $$x$$.

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