Low Points of a Sequence

For a finite sequence $E \colon a_1, a_2, \dots, a_M$ (with $M \geqslant 2$), an index $n \geqslant 2$ is called a low point of $E$ if $a_i > a_n$ for every $i = 1, 2, \dots, n-1$. If $E$ has exactly $p$ low points, we say $E$ is a sequence of order $p$.

1. Write down all low points of the sequence $E \colon -1, 2, 0, -2, 4, -5$. 2. Prove that if $E$ has order $0$, then $a_1 \leqslant a_n$ for $n = 1, 2, \dots, M$. 3. Suppose $E$ has order $k$ and satisfies $a_{n+1} \geqslant a_n - 2$ for all applicable $n$. Prove that