Writing a monotone function as the sum of two monotone functions

Question

Let $\mathbb{N}\equiv \{1,2,3,...\}$. Could you suggest two functions $f_1: \mathbb{N}\rightarrow \mathbb{R}$, $f_2: \mathbb{N}\rightarrow \mathbb{R}$ such that

$f_1$ is monotone decreasing,

$f_2$ is monotone increasing, and

$f\equiv f_1+f_2$ is monotone decreasing.

I managed to find $f_1, f_2$ such that $f\equiv f_1+f_2$ is monotone increasing (e.g., $f_1(n)=n^2$, $f_2(n)=1/n$) but not the other way around.

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Also, is there any theorem saying that any monotone (increasing or decreasing) function $f$ can be written as as the sum of a monotone increasing function $f_1$ and a monotone decreasing function $f_2$?

Answer 1

Let $f$ denote any monotone decreasing function from $\mathbb N\to \mathbb R$. Then let $f_1=2f$, $f_2=-f$. And similarly if $f$ is monotone increasing.

Taking, say, $f(n)=-n$ gives a particular example, but the general construction establishes the theorem you requested.

Answer 2

Let $f_1(x)$ have slope $m$. The general idea is to have $f_2(x)$ increasing but have a slope less than $|m|$, so that overall $f_1(x) + f_2(x)$ is decreasing.

You can do this with straight lines: Let $f_1(x) = -x$, and $f_2(x)$ be $\frac{1}{2}x$. Then $f_1(x)+f_2(x) = -\frac{1}{2}x$. Here's an illustration of this on Desmos.