Let a_1\geq \cdots \geq a_n \geq a_{n \plus{} 1} \equal{} 0 be real numbers. Show that
\sqrt {\sum_{k \equal{} 1}^n a_k} \leq \sum_{k \equal{} 1}^n \sqrt k (\sqrt {a_k} \minus{} \sqrt {a_{k \plus{} 1}}).
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