I'm wondering whether the following statement is true: Let $p$ and $q$ be two prime numbers (or more generally let $p$ and $q\neq 0$ be integers with $\gcd(p,q)=1$). Then for all $\varepsilon >0$ there exist an $n$ and $m$ in $\mathbb{N}$ such that $$\Bigg|\frac{p^n}{q^m}-1\Bigg|<\varepsilon.$$
Intuitively at least I'm convinced this should be true. Also experimentally it seems to hold. Is this a known result? If so, hints on how to show this are welcome.
Asking whether for all $\varepsilon>0$ there are $m,n$ such that $\left|\frac{p^n}{q^m}-1\right|<\varepsilon$ is equivalent to asking whether for all $\varepsilon>0$ there are $m,n$ such that $\left|n\log(p) - m\log(q)\right|<\varepsilon$, or, if we prefer, such that $\left|\frac{\log(p)}{\log(q)} - \frac{m}{n}\right|<\frac{\varepsilon}{n}$ (the $\varepsilon$ is different in all three cases, but the $\forall\varepsilon$ statements are equivalent). This, in turn, follows from the irrationality of $\frac{\log(p)}{\log(q)}$ and a well-known theorem by Dirichlet (or a standard fact on continued fractions) on approximation of irrationals by rationals.
(I'm just sketching the proof since you were asking for a hint.)