Problem: Prove that all positive rational numbers can be expressed as the finite sum of different numbers $\displaystyle \frac {1} {n}$ ($n$ is a natural number).
Example: $\displaystyle \frac {19}{16}=1+ \frac {1}{8} + \frac {1}{16}.$
*We cant sum numbers as $\displaystyle \frac {3}{16}$ (denominator > 1) but we can sum $\displaystyle \frac {1}{8}+ \frac {1}{16}.$
Any solutions? Suggestions?