Consider an arbitrary function of two variables $f \colon \mathbb N \times \mathbb R^+ \to \mathbb R$. My question is very simple: what does one mean by writing that
$$ \lim_{N \to \infty} \, f(N,r) = 0 $$
uniformly in $r \in (0,R]$, where $R > 0$ is an arbitrary fixed constant ?
My intuition. Intuitively, I think that the condition I stated above is equivalent to saying that for every $\epsilon > 0$ there exists a constant $L \in \mathbb N$ such that for every $N \in \mathbb N$ and $r \in (0,R]$ such that $N > L$, then it follows that $|f(N,r)| < \epsilon$ but I am not sure.
Can anyone confirm if I am right?
Thanks for any help in advance.