Trying to understand the irrationality criterion with examples.

Question

If the irrationality criterion below is wrong feel free to edit this post.

The irrationality criterion, as far I understood is that if exists sequences $p_n,q_n (n=1,2,3,...)$ where $p_n,q_n$ are integers then the number $\alpha \neq p_n/q_n$ is irrational if \begin{align} \left| \alpha -\frac{p_n}{q_n}\right|<\frac{1}{q_n^{1+\delta}} \end{align} where $\delta>0$.

If we multiply everything by $q_n$ we have $|q_n\alpha-p_n|<1/q_n^\delta$ so assuming that $\alpha=r/s$ with both $r,s$ integers, we find a integer in the interval $(0,1)$ because $\delta>0$, which is a contradiction.

So we must find sequences that $p_n,q_n$ such that $p_n/q_n$ converges very quickly to $\alpha$. How does one create such sequences and how to find $\delta$ ? I guess they don't fall from the sky.

For example, how we prove that $e$ is irrational using this criterion?

$$\left| e -\frac{p_n}{q_n}\right|<\frac{1}{q_n^{1+\delta}}$$

Answer 1

$e$ is rather exceptional in that it has a closed-form continued fraction:

$$e = [2; 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8,\ldots]$$

The corresponding convergents $p_n/q_n$ are $$ 2, 3, 8/3, 11/4, 19/7, 87/32, 106/39, 193/71, 1264/465, 1457/536, \ldots $$ which satisfy your inequality with $\delta = 1$.

Of course, if you have the continued fraction you really don't need your criterion: any non-terminating simple continued fraction represents an irrational number.