# Convergence of rationals to irrationals and the corresponding denominators going to zero

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 $y$ ?

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Hint: For every positive integer $n$, consider the set $R(n)$ of rational numbers $p/q$ such that $1\leqslant q\leqslant n$. Show that for every $n$ the distance $\delta(n)$ of $y$ to $R(n)$, defined as $\delta(n)=\inf\{|y-r|\mid r\in R(n)\}$, is positive. Apply this to any sequence $(p_k/q_k)$ converging to $y$, showing that for every positive $n$, there exists $k(n)$ such that for every $k\geqslant k(n)$, $p_k/q_k$ is not in $R(n)$, hence $q_k\geqslant n+1$.