SCHWEITZER 2020 Problem 7

Let $p(n)\geq 0$ for all positive integers $n$ . Furthermore, $x(0)=0, v(0)=1$ , and $$x(n)=x(n-1)+v(n-1), v(n)=v(n-1)-p(n)x(n) (n=1,2,\dots).$$ Assume that $v(n)\to 0$ in a decreasing manner as $n \to \infty$ . Prove that the sequence $x(n)$ is bounded if and only if $\sum_{n=1}^{\infty}n\cdot p(n)<\infty$ .