Let $k$ be an integer greater than or equal to $1$, $(u_{0}, \ldots, u_{k})$ a finite sequence of integers, and $a$ an integer such that $a > u_{p}$ for all $p \in \llbracket 0, k \rrbracket$. We insert the value $a$ into this sequence immediately after $u_{i}$, with $i \in \llbracket 0, k-1 \rrbracket$, to obtain the sequence $(u_{0}, \ldots, u_{i}, a, u_{i+1}, \ldots, u_{k})$. Compare the number of ascents and descents of the new sequence with respect to the old one. Two cases will be distinguished.
Let $k$ be an integer greater than or equal to $1$, $(u_{0}, \ldots, u_{k})$ a finite sequence of integers, and $a$ an integer such that $a > u_{p}$ for all $p \in \llbracket 0, k \rrbracket$. We insert the value $a$ into this sequence immediately after $u_{i}$, with $i \in \llbracket 0, k-1 \rrbracket$, to obtain the sequence $(u_{0}, \ldots, u_{i}, a, u_{i+1}, \ldots, u_{k})$. Compare the number of ascents and descents of the new sequence with respect to the old one. Two cases will be distinguished.