Provide me an example of a sequence that has finite number of peaks.
Definition: Let $<a_n>$ be a sequence then $a_m$ is peak point or a peak of $<a_n>$ if $a_m \ge a_n$ $\forall n \ge m$
My Attempt:
I know an example of a sequence$$a_n = \left\{\begin{array}{lr} {1\over n} + 1 & \text{for } n \leq 5\\ n & \text{for } n >5 \end{array}\right.$$ that has finite number of peak points. But it does not satisfy definition of peaks. Please help me.
$\endgroup$ 21 Answer
$\begingroup$It is easy to construct such sequences based on an original bounded sequence which is monotone increasing. For instance, $a_n = \arctan(n)$ is one, bounded above by $\pi/2$. So you can specify particular $a_i$ such that $(a_i)$ is a finite monotone decreasing sequence and $a_i > \pi/2 \; \forall i$. Then each $a_i$ specified is a peak per the definition provided.
Example:
$$a_n = \left\{ \begin{matrix} 10 & n=1 \\ 9 & n=2 \\ 8 & n=3 \\ \arctan(n) & n \ge 4 \end{matrix} \right.$$
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