Definition of non-monotonic sequence

Question

I am working on other proofs, namely for so-called "quasi-increasing" but non-monotonic sequences.

But, my question: is the following a sufficient definition of non-monotonic?

For all $N \in \mathbb{N}$ where $l>n>m \ge N$ there is $a_n \le a_m$ and $a_l \ge a_n$

Namely, there are always terms with preceding and following terms both greater.

I am not sure if the quantification is correct, as in I do not want to say each element has all elements before and after it greater. This would seem contradictory to me, and misses the idea of mere non-monotonicity.

Answer 1

Negate the definition:

$\exists m,n, p, q \in\mathbb{N}$ s.t ($ n>m $ but $x_n<x_m$) and ($p>q$ but $x_p >x_q$)