« Vrati se
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}.

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