If $(\frac{p_k}{q_k})$ is a sequence of rationals that converges to an irrational $y$, how do you prove that $(q_k)$ must go to $\infty$?
I thought some argument along the lines of "breaking up the interval $(0,p_k)$ into $q_k$ parts", but I'm not sure how to put it all together. Perhaps for every $q_k$, there is a bound on how close $\frac{p_k}{q_k}$ can get to the irrational $y$ ?