Let $\, k_1 < k_2 < k_3 < \cdots \,$ be positive integers, no two consecutive, and let $\, s_m = k_1 + k_2 + \cdots + k_m \,$ for $\, m = 1,2,3, \ldots \; \;$. Prove that, for each positive integer $\, n, \,$ the interval $\, [s_n, s_{n+1}) \,$ contains at least one perfect square.