Let $x$ be a positive rational number, strictly between $0$ and $1$. Prove that there is a finite strictly increasing list of positive integers $2 \leq q_1<q_2<\cdots<q_N $ such that $$x={1\over q_1}+{1\over q_2}+ \cdots+{1\over q_N}.$$
I have tried many methods, such as mathematical induction. I know it is obvious that $\frac{1}{m}$ is right for assumption, but when I begin to prove $\frac{2}{m}$, I feel it is hard to find some patterns for $\frac{2}{m}$. Thus, maybe my thoughts are wrong.
Also, I still tried to change the fraction so that the numerator will be smaller and smaller. But I still cannot find a way to lower the numerator. Can someone help me solve the question? Or, can someone give me some hints. I will appreciate you very much!
Let $q=m/n$ rational, with $q\in (0,1)$, and hence $$ \frac{1}{k}\le\frac{m}{n}<\frac{1}{k-1} $$ for some integer $k>1$. If $\frac{1}{k}=\frac{m}{n}$, we have nothing to prove. Otherwise, we have $$ \frac{m}{n}-\frac{1}{k}=\frac{km-n}{kn}=\frac{m'}{n'}>0. $$ But $\dfrac{1}{k-1}>\dfrac{m}{n}$ implies that $n>m(k-1)$, or $m>km-n=m'>0$.
Hence, after we subtracted the the nearest and smallest fraction of the form $1/k$, the remainder $m'/n'$ has smaller numerator.
Hence, in at most $m$ steps this procedure ends, and thus we obtain the sought for representation.