ISL 2009 A6

Suppose that $s_{1}, s_{2}, s_{3}, \ldots$ is a strictly increasing sequence of positive integers such that the subsequences $$s_{s_{1}}, s_{s_{2}}, s_{s_{3}}, \ldots \text { and } s_{s_{1}+1}, s_{s_{2}+1}, s_{s_{3}+1}, \ldots$$ are both arithmetic progressions. Prove that $s_{1}, s_{2}, s_{3}, \ldots$ is itself an arithmetic progression.