RO-TST 1988 Problem 4

Prove that for all positive integers $0<a_1<a_2<\cdots <a_n$ the following inequality holds: $$(a_1+a_2+\cdots + a_n)^2 \leq a_1^3+a_2^3 + \cdots + a_n^3 .$$ Viorel Vajaitu