ISL 1982 Problem 28

Let $(u_1, \ldots, u_n)$ be an ordered $n$ tuple. For each $k, 1 \leq k \leq n$ , define $v_k=\sqrt[k]{u_1u_2 \cdots u_k}$ . Prove that $$\sum_{k=1}^n v_k \leq e \cdot \sum_{k=1}^n u_k.$$ ( $e$ is the base of the natural logarithm).