Problem 3

Suppose that

s_1, s_2, s_3, \dots

is a strictly increasing sequence of positive integers such that the subsequences

s_{s_1}, s_{s_2}, s_{s_3}, \dots

and

s_{s_1+1}, s_{s_2+1}, s_{s_3+1}, \dots

are both arithmetic progressions. Prove that the sequence

s_1, s_2, s_3, \dots

is itself an arithmetic progression.

Step 5 of 5: The increment is forced to be constant

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In plain words

Once the largest and smallest increments coincide, every increment in between must equal that common value too.

M=m=B-A \implies d_n \equiv B-A \implies (s_n)\text{ is arithmetic}

Detailed analysis

Steps 3 and 4 give $M=B-A=m$ . Since $m\le d_n\le M$ for every $n$ and $m=M$ , every $d_n$ equals this common value. So $s_{n+1}-s_n$ is constant, meaning $s_1,s_2,s_3,\dots$ is an arithmetic progression, as required. $\blacksquare$