The irrationality measure can be defined as:
Let $x$ be a real number, and let $R$ be the set of positive real numbers $\mu$ for which
$$0<|x-\frac{p}q|<\frac1{q^\mu}$$
has (at most) finitely many solutions for $p$ and $q$ integers. Then, $$\mu(x)=\text{inf}_{\mu\in R}\, \mu$$
My thoughts are:
Suppose for some $x$, $(\mu_0,p_0,q_0)$ is a solution to the inequality, then because $$\frac1{q_0^{\mu_0}}<\frac1{q_0^{\mu_0-\epsilon}}$$ for every $\epsilon>0$, thus $(\mu_0-\epsilon,p_0, q_0)$ is also a solution to the inequality. Thus, we can reduce the irrationality measure as small as possible.
What’s the flaw in my thought?
The point of the definition is that there are at most finitely many solutions. You've only ensured that all the solutions are still solutions; you didn't consider whether you've created more solutions. In fact, $\mu$ is defined precisely such that if you reduce it, you get infinitely many solutions.