Is it true that every positive rational number $r = \frac{n}{m}$ could be represented as sum of $\frac{1}{k}$ for different $k$'s:

$$ r = \sum_{i=1}^{s} \frac{1}{c_i} $$ where all $c_i$ are different?

If true, are there any bounds on number of summands $s$ based on $n$ and $m$?

greedy algorithmalways works here: at each stage, choose $\frac{1}{c_i}$ to be as large as possible so that the partial sum is less than or equal to $r$. – Pete L. Clark Jul 22 '10 at 14:03