I have question regarding Kinchin's Theorem. Part of the Theorems says the following
**[Kinchin's Theorem] ** Let $p,q\in \mathbb{N}$ and $a \in \mathbb{R}$. Then if $\sum_{z_x=1}^\infty qf(q)$ diverges where $f(q)$ is nonnegative function, for every constant $c>0$ and almost every $a \in \mathbb{R}$ there are infinitely many values of $q$ such that \begin{align} \min_{p \in \mathbb{N}} |a-\frac{p}{q}| \le c f(q) \end{align}
My question is the following. If I restrict values of $p$, say $p \le N$. So, the that we have \begin{align} \min_{p \in \mathbb{N}: p \le N} |a-\frac{p}{q}| \end{align}
What can we say about this case?
Clearly the values of q should be finite now, right?
Also, can we even have a bound of the form
\begin{align} \min_{p \in \mathbb{N}: p } |a-\frac{p}{q}| \le c f(q) \end{align}
I am sorry if I am being not clear on something I am not a number theorist. Thank you very much.