MathNet 2013 Question 52

Let $n$ be a positive integer and $a_1,\ldots,a_n$ real numbers such that $a_1+\cdots+a_k\leq k$ for every $k\in\{1,2,\ldots,n\}$. Prove that
$$\frac{a_1}{1}+\frac{a_2}{2}+\cdots+\frac{a_n}{n}\leq\frac11+\frac12+\cdots+\frac1n.$$