Školjka

%V0 We define a sequence $\left(a_{1},a_{2},a_{3},...\right)$ by setting $$a_{n} = \frac {1}{n}\left(\left[\frac {n}{1}\right] + \left[\frac {n}{2}\right] + \cdots + \left[\frac {n}{n}\right]\right)$$ for every positive integer $n$. Hereby, for every real $x$, we denote by $\left[x\right]$ the integral part of $x$ (this is the greatest integer which is $\leq x$). a) Prove that there is an infinite number of positive integers $n$ such that $a_{n + 1} > a_{n}$. b) Prove that there is an infinite number of positive integers $n$ such that $a_{n + 1} < a_{n}$.