Prove by induction that the equation $\dfrac{1}{x_1} + \dfrac{1}{x_2} + ... + \dfrac{1}{x_n} = 1$ has a solution with $x_1<x_2<...<x_n$.
For every $n\ge3$ there exists $n$ different integers $a_1,...,a_n$ so that $\frac{1}{a_1}+\frac{1}{a_2}+...+\frac{1}{a_n}=1$