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I’m struggling with a problem that reads pretty much as follows:

Prove that every positive rational number $r$ can be represented as a finite sum of rational numbers of the form $\frac{1}{k}$ where every $k$ is distinct. I believe $k$ is a natural number, but this isn’t explicitly stated in the exercise.

I would like a tip on how to start or a small clue that would be useful to begin solving the problem.

Bill Dubuque
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TheTopologicalEggs39
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  • Well, if you don't require $k\in \mathbb N$ you can just take $k=\frac 1r$. And, usually the rule is that the $k's$ should be distinct (otherwise the problem is trivial). – lulu Oct 27 '23 at 13:47
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    Well, $\frac{a}{b}$ can trivially be written as $\frac{1}{b}+\cdots+\frac{1}{b}$ with $a$ times $\frac{1}{b}$ , the problem is more interesting if we demand that the $k$'s are distinct. Even then , there is always a solution as the theory of egyptian fractions tells us. The greedy algorithm does the job. – Peter Oct 27 '23 at 13:48
  • @Paul you can also convert them to distinct ones – Benjamin Wang Oct 27 '23 at 13:49
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    I forgot to clarify at the beginning that every $k$ has to be distinct. – TheTopologicalEggs39 Oct 27 '23 at 13:50

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