I really don't understand how to do proofs on convergence at ALL.
I know you're supposed to use
$|x_i – x| < \epsilon$
but I have no idea how to apply this to this question:
Show that if $x$ is any real number then there is an infinite sequence of rational numbers converging to $x$.
Any hints or help will be greatly appreciated!
Best Answer
Are you familiar with the density of the rationals? If so,
For every $n\in \mathbb{N}$, let $x_n$ be a rational in the open ball $B(x,\frac{1}{n})$. By the density of the rationals, there is always one such a $x_n\in\mathbb{Q}$. Then for any $\epsilon>0$, there exists $N$ such that $\frac{1}{N}<\epsilon$ by the Archimedean property. Then, for all $n>N$, $x_n \in B(x,\frac{1}{N})\subset B(x,\epsilon),$ meaning $|x_n -x|< \epsilon$.