Analysis, Density of Rational Numbers

Question

Suppose $p/q$ and $k/l$ are rational numbers with $\left|(p/q - k/l)\right|< {1}/{ql}$. Prove $p/q = k/l$.

Similarly, let $p/q$ be a fixed rational number and suppose $k/l$ is a rational number with $0 < \left|p/q - k/l \right| < 1/qm$ ... for some natural number m. Show that $m < l$.

The work I've done with these is just algebra to split up the absolute value inequalities but I can't logically work out the problems. I'd prefer tips over a complete answer to the problem as these are for homework.

Answer 1

Hint: $pl-qk \in \mathbb{Z}$.

$a \in \mathbb{Z},\ |a|<1 \quad \Longrightarrow \quad a = 0$.