VN-TST 1999 Problem 4

Let a sequence of positive reals $\{u_n\}^{\infty}_{n=1}$ be given. For every positive integer $n$ , let $k_n$ be the least positive integer satisfying:

$$\sum^{k_n}_{i=1} \frac{1}{i} \geq \sum^n_{i=1} u_i.$$

Show that the sequence $\left\{\frac{k_{n+1}}{k_n}\right\}$ has finite limit if and only if $\{u_n\}$ does.