In George Lowther's CW answer taking care of the "simple case" there are a Lemma 1, 2 and 3 concerning sets of function $U$ and $V$ about which it is stated "they could possibly be standard results, but I've never seen them before." I wanted to add that, by performing an inversion, those results become instead statements about rapidly decaying functions and Schwartz functions. In the latter form, they are easier to find in the literature. Really the present answer would be better as a comment, but I think there will not be space for that.

Use the map $x \mapsto 1/x$ to put functions on $(0,1]$ into bijection with functions on $[1,\infty)$. This correspondence puts the functions $f$ on $(0,1]$ with $\lim_{x \to 0^+} x^{-n} f(x) = 0$ for all $n$ into bijection with the functions $f$ on $[1,\infty)$ with $\lim_{x \to \infty} f(x) x^n = 0$ for all $n$, i.e. the functions of rapid decay. It also puts the functions $f$ on $(0,1]$ for which putting $f(x)=0$ for $x\leq 0$ yields a smooth extension into bijection with the Schwartz functions on $[1,\infty)$. 

So George Lowther's Lemma 1 is equivalent to asking whether, for every function which decays rapidly at $\infty$, there exists a Schwartz function which decays more slowly. Lemmas 2 and 3 can be given similar restatements. Stated in this revised form, Lemma 1 is arguably easier to prove. 

>**Claim:** Suppose $f$ is a rapidly decaying function on $[1,\infty)$, then there exists a Schwartz function $g$ on $[1,\infty)$ with $g \geq |f|$.

>**Proof:** Without loss of generality, $f$ is postive-valued and decreasing, or else replace it by $x \mapsto \sup_{y \geq x} f(y)$. Let $\varphi$ be a nonnegative $C^\infty$ bump function with support in $[0,1]$ and $\int_\mathbb{R} \varphi =1$. Then it is simple to check that the convolution $g= \varphi * f$ is a Schwartz function with $g \geq f$ (the main point is that rapidly decaying functions are closed under convolution, which implies the convolution of a Schwartz function and a rapidly decaying function is Schwartz). 

The Schwartz VS rapid decay versions of the facts seem to be easier to find in the literature as well. See this expository note of Paul Garret: [Schwartz-function envelopes for rapidly decreasing functions](https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&ved=2ahUKEwiBm8KExdjjAhXHnOAKHU3rCXEQFjAAegQIAxAC&url=http%3A%2F%2Fwww-users.math.umn.edu%2F~garrett%2Fm%2Ffun%2Fweil_schwartz_envelope.pdf&usg=AOvVaw32IkcehhK0xxXzWlP8qeP_) and note that the three bullet points of the theorem in Garrett's note correspond more or less to the three Lemmas in Lowther's answer. This [post](https://math.stackexchange.com/questions/2175092/given-a-schwartz-function-is-it-always-possible-to-write-it-as-a-product-of-two) by Abdelmalek Abdesselam points to additional references in the literature.

- K. Miyazaki, [Distinguished elements in a space of distributions](https://projecteuclid.org/euclid.hmj/1555615830), Lemma 1.
-  H. Petzeltová; P. Vrbová [Factorization in the algebra of rapidly decreasing functions on $R_n$](https://eudml.org/doc/16918).
- J. Voigt, [Factorization in some Frechet algebras of differentiable functions](https://eudml.org/doc/218540).