This question is an offshoot from this previous MSE post.

I have a ratio of two numbers $a$ and $b$ (presumably both positive integers), where $a$ and $b$ are determined by some arithmetic / number-theoretic equation $f(a,b)=0$.

Let $S$ be the set $$S = \{(a,b)|f(a,b)=0 \}.$$

Suppose that I have bounds for the sum $$L(a,b) \leq \frac{a}{b} + \frac{b}{a} \leq U(a,b).$$

If I want to prove that $S$ is empty, one way is to prove that the ratio $a/b$ (or equivalently, $b/a$) is actually irrational.

Since we know that, in general $$\frac{a}{b} + \frac{b}{a} \rightarrow \infty,$$ does the upper bound $U(a,b)$ guarantee an irrationality proof?

If anybody can point out to me a proof (in Diophantine approximation) that proceeds along similar lines, I will most certainly appreciate it.