Suppose that $\{x_j:j\ge1\}$ are positive real numbers such that $\sum_{j=1}^\infty1/x_j<\infty$ and $x_i\ne x_j$ when $i\ne j$. Define the product $$ P_i^n=\prod_{j=1,j\ne i}^n\biggl(1-\frac{x_i}{x_j}\biggr). $$ Does this product converge as $n\to\infty$?

Under an additional assumption of the monotonicity of the sequence $\{x_i:i\ge1\}$, I am able to prove the convergence. I show that the logarithm of the sequence is bounded using an inequality for logarithms. However, without the assumption of the monotonicity, this does not work. I am not even sure that this product converges.

Any help is much appreciated!


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