Definition $1.$ An irrational number $x$ is called a Liouville number if for any positive integer $n$ there exists a pair of integers $(p,q)$ such that $q\gt 1$ and $|x-p/q|\lt 1/q^n$.
Definition $2.$ An irrational number $x$ is called a Liouville number if for any positive integers $N,n$ there exists a rational number $p/q$ such that $|x-p/q|\lt \frac{1}{Nq^n}$.
I came across these definitions from different sources. I want to prove the Liouville’s theorem using definition $2$ while the proof that I know assumes Definition $1$.
Are these two definitions equivalent? If not, what if in Definition $2$ we add the requirement that $q\gt 1$?
The definitions are equivalent. If $x$ satisfies Definition 2, then by choosing $N$ such that $1/N<|x-m|$ for all integer $m$, we see that $q>1$, so Definition 1 holds for $x$ as well.
Conversely, If $x$ satisfies Definition 1, and we want to verify Definition 2 for a given $N,n$, we just apply Definition 1 with $n*=N+n$ in place of $n$: There exist integers $p$ and $q>1$ such that $$|x-p/q|<q^{-N-n}<\frac1{Nq^n}\,,$$ where we used the inequality $q^N \ge 2^N >N$.