What are the applications of proving some results related to number theory like the one given below?

Question

I had this problem in my Mathematics textbook. Here's how it goes : $$\text{For any two positive integers, } a \text{ and } b \text{, prove that } \sqrt{2} \text{ always lies between } \dfrac{a}{b} \text{ and } \dfrac{a+2b}{a+b}$$ I did prove this by taking two cases :

$\dfrac{a}{b} > \dfrac{a+2b}{a+b}$

$\dfrac{a+2b}{a+b} > \dfrac{a}{b}$
and then obtaining the following results respectively :

$\dfrac{a}{b} < \sqrt{2} < \dfrac{a+2b}{a+b}$

$\dfrac{a+2b}{a+b} < \sqrt{2} < \dfrac{a}{b}$

But, I couldn't help but wonder about the applications of these proofs. I do know that a lot of pure mathematics, number theory, to be specific, has applications in the field of Computer Science and Physics.
I think that such proofs might also be used to derive some property that when used in a separate problem, yields something with practical applications in (maybe) Engineering, Science or some other Mathematical fields.

I'd like to know about some applications of these types of proofs. Some examples would be appreciated.

Thanks!