MathDB

We define a sequence

\left(a_{1},a_{2},a_{3},\ldots \right)

by a_{n} \equal{} \frac {1}{n}\left(\left\lfloor\frac {n}{1}\right\rfloor \plus{} \left\lfloor\frac {n}{2}\right\rfloor \plus{} \cdots \plus{} \left\lfloor\frac {n}{n}\right\rfloor\right), where

\lfloor x\rfloor

denotes the integer part of

x

.
a) Prove that

a_{n+1}>a_n

infinitely often. b) Prove that

a_{n+1}<a_n

infinitely often.
Proposed by Johan Meyer, South Africa

Problem Statement