We define a sequence
by setting
for every positive integer
. Hereby, for every real
, we denote by
the integral part of
(this is the greatest integer which is
).
a) Prove that there is an infinite number of positive integers
such that
.
b) Prove that there is an infinite number of positive integers
such that
.
![\left(a_{1},a_{2},a_{3},...\right)](/media/m/c/e/8/ce854648a859e5694c58621a66fe6f87.png)
![a_{n} = \frac {1}{n}\left(\left[\frac {n}{1}\right] + \left[\frac {n}{2}\right] + \cdots + \left[\frac {n}{n}\right]\right)](/media/m/0/f/c/0fcd7236589e4454f518b8d8aa9fe147.png)
for every positive integer
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
![x](/media/m/f/1/8/f185adeed9bd346bc960bca0147d7aae.png)
![\left[x\right]](/media/m/3/6/9/3697c66f8530757a1166f24a1fd325e6.png)
![x](/media/m/f/1/8/f185adeed9bd346bc960bca0147d7aae.png)
![\leq x](/media/m/2/7/3/27302824e7ae9e57b6a2d2e770bdfc1f.png)
a) Prove that there is an infinite number of positive integers
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
![a_{n + 1} > a_{n}](/media/m/e/f/b/efb49f0c689c701d59d06184a2ff714f.png)
b) Prove that there is an infinite number of positive integers
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
![a_{n + 1} < a_{n}](/media/m/5/6/e/56ea4fc20888476c227c3b3b80c9a64d.png)