Ordinals and recursive algorithms

We prove the following result, which generalizes results of Erickson et al. (2022) and Bufetov et al. (2024): Call a function $f:\mathbb{R}\to \mathbb{R}$ ordinal decreasing if for every infinite decreasing sequence $x_0>x_1>x_2>\cdots$ there exist $i<j$ such that $f(x_j) \geq f(x_i)$. Given ordinal decreasing functions $f,g_0,\ldots,g_k,s$ that are larger than $0$, define the recursive algorithm "$M(x)$: if $x<0$ return $f(x)$, else return $g_0(-M(x-g_1(-M(x-\cdots-g_k(-M(x-s(x)))\cdots))))$". Then $M(x)$ halts and is ordinal decreasing for all $x \in \mathbb{R}$.