From Vladimir Zorich Analysis I:
Let us call an irrational number $a \in \mathbb{R}$ well approximated by rational numbers if for any natural number $n, N \in \mathbb{N}$ there exists a rational number $\frac{p}{q}$ such that $\lvert a - \frac{p}{q} \rvert < \frac{1}{Nq^n}$.
(a) Construct an example of a well-approximated irrational number.
(b) Prove that a well-approximated irrational number cannot be algebraic, that is, it is transcendental (Liouville's theorem).
The answer to the question (a) is $a=\sum_{n \geq 1}{\frac{1}{10^{n!}}}$, where $a$ is called Liouville's constant.
As for (b):
I've got stuck almost immediately due to lack of my knowledge of number theory. I have to show for any polynomial $P(x)=\sum_{k=0}^n{a_{k}x^k}$:
$$\forall l \in \mathbb{R}, \lvert l - \frac{p}{q} \rvert < \frac{1}{Nq^n}, P(l) \neq 0$$
But suppose:
$$P(l)=0$$
Then we can assume few points:
- $P(x)$ might have a set of rational roots $S \subset \mathbb{Q}$, such that $l \notin S$
- There exists $\frac{p}{q} \notin S$ such that $P(\frac{p}{q}) \neq 0$
- $P(\frac{p}{q})=\sum_{k=0}^na_k\frac{p^k}{q^k}=a_1 \frac{p}{q} + a_{2} \frac{p^2}{p^2} + ... + a_{k} \frac{p^k}{q^k}$. Therefore it is seen that $P(\frac{p}{q})=\frac{1}{q^n}c$ for some $c \in \mathbb{R}$ and thus $P(\frac{p}{q}) \geq \frac{1}{q^n}$
- Since $P(x)=0$, then $\lvert P(x) - P(\frac{p}{q}) \rvert \geq \frac{1}{q^n}$
- By triangle inequality, $\lvert l - \frac{p}{q} \rvert < \frac{1}{Nq^n} \implies \lvert l \rvert + \lvert \frac{p}{q} \rvert < \frac{1}{Nq^n} \implies \frac{p}{q} < \frac{1}{Nq^n} + \lvert l \rvert$
Question:
Can I utilize the points above to construct a simple elegant proof for Liouville's theorem? Perhaps constraints from rational root theorem can be used in some way?
Thank you!
By the way: I've been able to find a single proof from small research, but I find it slightly implicit and complex and struggle to understand the essence of some parts.
Also, I do wonder what kind of solution would be most suitable from real-analysis perspective, since Zorich only briefly touched definition of transcendal numbers and $q$-nary approximations.