61. (SWE 3) Let $a_{1} \leq a_{2} \leq \cdots \leq a_{n}$ and $b_{1} \leq b_{2} \leq \cdots \leq b_{n}$ be two sequences such that $\sum_{k=1}^{m} a_{k} \geq \sum_{k=1}^{m} b_{k}$ for all $m \leq n$ with equality for $m=n$. Let $f$ be a convex function defined on the real numbers. Prove that
$$\sum_{k=1}^{n} f\left(a_{k}\right) \leq \sum_{k=1}^{n} f\left(b_{k}\right)$$