Borel’s theorem, in its simplest form, states that for any sequence $(a_k)$ of real numbers, there is a function $f\in {\cal C}^{\infty}({\mathbb R},{\mathbb R})$ such that $f^{(k)}(0)=a_k$ for all $k$. I wonder if we can take $f$ to be increasing.
Obviously, this will not be possible if the sequence starts with some zeros and then a negative number. If, however, all the $a_k$ are $0$ or if the first nonzero $a_k$ is positive, there seems to be no reason why $f$ could not be increasing. Is this already known ?