real analysis - Dependencies in 'epsilon of room'-style proofs

I want to try and prove that

If $\{a_n\}_{n=1}^\infty$ is a positive sequence converging to zero, then there exists $N \in \mathbb{N}$ such that $a_{n+N} \leq a_n$ for every $n \in \mathbb{N}$.

Proof. Given $\varepsilon > 0$, it was easy to prove that there exists $N$ that depends on $\varepsilon$, such that $a_{n+N} \leq a_n + \varepsilon$. For as $\{\sup_{n \geq k}a_n\}_{k=1}^\infty$ is decreasing, I pick $N$ such that $\sup_{n \geq k} a_n < \varepsilon$ whenever $k \geq N$. This means that for every $n \in \mathbb{N}$, $a_{n+N} < \varepsilon \leq \varepsilon + a_n$.

Question: I have heard of 'Epsilon of room'-proofs, but I cannot find out whether I am allowed to let $N$ depend on $\varepsilon$ in such proofs, or not. Could you provide and answer to whether such a dependecy is allowed, and also explain what kind of dependencies are allowed and forbidden in such proofs.