Question 20
Let a positive integer $d$ and a sequence $\left(a_{i}\right)_{i=1,2,3, \ldots}$ of positive integers be given. Assume that the following two conditions are satisfied:
- Every positive integer occurs exactly once in the sequence.
- For all indices $i \geq 10^{100}$ we have $\left|a_{i+1}-a_{i}\right| \leq 2 d$.
Prove that there exist infinitely many indices $j$ for which $\left|a_{j}-j\right|\lt d$.
- Every positive integer occurs exactly once in the sequence.
- For all indices $i \geq 10^{100}$ we have $\left|a_{i+1}-a_{i}\right| \leq 2 d$.
Prove that there exist infinitely many indices $j$ for which $\left|a_{j}-j\right|\lt d$.