ISL 1997 Problem 19

Let $a_1\geq \cdots \geq a_n \geq a_{n + 1} = 0$ be real numbers. Show that $$\sqrt {\sum_{k = 1}^n a_k} \leq \sum_{k = 1}^n \sqrt k (\sqrt {a_k} - \sqrt {a_{k + 1}}).$$ Proposed by Romania