There occurs no $k$ in the definition of $S_k$

Actually, it is $$S_k=\sum_{A\subset \{x_1,x_2,...,x_k\}} \frac{1}{\prod_{a\in A}a} , k=1,2,...,n.$$

Suppse that $n\geq 2$ and $0<x_1<x_2<...<x_n$ are integer numbers. We denote that : $$S_k=\sum_{A\subset \{x_1,x_2,...,x_k\}} \frac{1}{\prod_{a\in A}a} , k=1,2,...,n.$$ (where $A$ is a non-empty subset). Show that if $S_n ,S_{n-1}$ were positive integer numbers , then $\forall k : S_k$ is a positive integer.

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