2

Part of 2018 Germany Team Selection Test

Problems(1)

Permutation of Positive Integers with Bounded Difference

Source: Germany VAIMO 2018 P2

A positive integer

d

and a permutation of positive integers

a_1,a_2,a_3,\dots

is given such that for all indices

i\geq 10^{100}

|a_{i+1}-a_{i}|\leq 2d

holds. Prove that there exists infinity many indices

j

|a_j -j|< d