Suppose $(x_n)_n$ is a non-decreasing sequence of natural numbers. Then, $\sum\frac{1}{x_{x_n}-x_n}$ diverges $\iff\sum\frac{1}{x_n}$ diverges.

Question

Is the following proposition true, and if so, how can it be proven? I cannot find a counter-example.

Proposition: Suppose $\ (x_n)_n\ $ is a non-decreasing sequence of natural numbers, and $\ x_{x_n} \neq x_n\ \forall n\in\mathbb{N}.\ $ Then, $\ \displaystyle\sum_n \frac{1}{x_{x_n} - x_n}\ $ diverges $\ \iff \displaystyle\sum_n \frac{1}{x_n}\ $ diverges.

Note: if we did not require $\ (x_n)_n\ $ to be non-decreasing, then $\ 2, 10, 4, 100, 6, 1000, 8, 10000, 12, 100000,\ldots\ $ would be a counterexample.

Answer 1

It seems that $\ x_n = \left\lfloor (n+1)\log(n+1)\right\rfloor $ is a counterexample.