Prove that
\begin{equation*} \frac{1}{x_1}+\frac{1}{x_2}+\frac{1}{x_3}+\cdots +\frac{1}{x_n}=1,~∀i,x_i\in \mathbb{Z^+} \end{equation*}
has a finite number of integer solutions. I tried to solve $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=1$ in integers and tried to deduce a solution for $\frac{1}{x}+\frac{1}{y}=1.$ And apply induction with the induction step that the number of solutions is finite from $n=1$ to $m-1$ and from there construct a solution for $n=m.$ I found
\begin{equation*} \frac{1}{(z-1)x}+\frac{1}{(z-1)y}= \frac{1}{z} \end{equation*}
but I rather had $1$ there in stead of $\frac{1}{z}.$ Stuck now. Help impossible for me?