Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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 $y$ ?

share|improve this question
add comment

1 Answer

up vote 1 down vote accepted

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 $q_k\geqslant n+1$.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.