I am trying to show that for any real number a, there exist infinitely many rational numbers m/n with $ |a – m/n| < 1 /n^{2} $. I've tried to attempt the question by assuming there are finite rational numbers and finding a contradiction.
[Math] Proving that rational numbers are dense
analysisrational numbers
Best Answer
Try to show that for every real number $a$ and every positive integer $N$ there exist $p,q$ integers with $1 \le q \le N$ such that $|qa - p| \le 1/(N+1)$.
This can be shown just using the pigeonhole principle in a clever way.
The result is called Dirchlet's approximation theorem. In this way you could easily find more detailed information in case it is needed.