MathDB

Consider the following sequence of positive real numbers

\dots<a_{-2}<a_{-1}<a_0<a_1<a_2<\dots

infinite in both directions. For each positive integer

k

let

b_k

be the least integer such that the ratio between the sum of

k

consecutive terms and the greatest of these

k

terms is less than or equal to

b_k

(This fact occurs for any sequence of

k

consecutive numbers). Prove that the sequence

b_1,b_2,b_3,...

coincides with the sequence

1,2,3,...

or is eventually constant.

Problem Statement