A question about how to express a fraction as $\frac{1}{{q}_{1}}+\frac{1}{{q}_{2}}+\cdots +\frac{1}{{q}_{N}}$

Let x be a positive rational number, strictly between 0 and 1. Prove that there is a finite strictly increasing list of positive integers $2\le {q}_{1}<{q}_{2}<\cdots <{q}_{N}$ such that

$x=\frac{1}{{q}_{1}}+\frac{1}{{q}_{2}}+\cdots +\frac{1}{{q}_{N}}.$

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!