Question 4
Let $(u_n)$ be a sequence of positive real numbers. For each positive integer $n$, let $k_n$ be the least positive integer satisfying $\sum_{i=1}^{k_n}1/i\ge\sum_{i=1}^n u_i$. Prove that $(k_{n+1}/k_n)$ has a finite limit if and only if $(u_n)$ has a finite limit.