For a positive real number $c>0$,call a positive integer $c$-$good$ if for all positive integer $m<n$,$\frac {m} {n}$ can be represented as

$\frac{m}{n}=\frac{a_{0}}{b_{0}}+...+\frac{a_{k}} {b_{k}}$

for some non-negative integers $k<\frac{n} {c}$ and $2b_{i}\leq n$ and $0\leq a_{i}<b_{j}$ and $0\leq j\leq k$.Show that,for any real $c$,there are an infinite number of $c$-$good$ positive integers.

## Good positive integers

### Good positive integers

"Questions we can't answer are far better than answers we can't question"

### Re: Good positive integers

Aha. That problem of mine! It's quite easy actually. But it can be made a bit hard with more restrictions. Have you solved it?

One one thing is neutral in the universe, that is $0$.