Borel's lemma on prescribed derivatives

Prove that if $n$ is a positive integer and $\alpha , \varepsilon$ are real numbers with $\varepsilon > 0$, then there is a real function $f$ with derivatives of all orders such that

1. $\left|f^{(k)}(x)\right| \leq \varepsilon$ for $k = 0, 1, \ldots , n-1$ and all $x \in \mathbb {R}$,

2. $f^{(k)}(0) = 0$ for $k = 0, 1, \ldots , n-1$,

3. $f^{(n)}(0) = \alpha$.

Prove that for any numerical sequence $\left\{ a_{n}\right\}_{0}^{\infty }$ there exists a function $\varphi \in \mathcal{D}$ such that $\varphi^{(n)}(0) = a_{n}$ for all $n = 0,1,2, \ldots$

Let $\varphi \in C^{\infty }[a, b]$ (at the endpoints of the segment the existence of the corresponding one-sided derivatives of every order is required). Prove that there exists a function $\psi \in \mathcal{D}$ such that $\left.\psi \right|_{[a, b]} \equiv \varphi$.

Let $\left\{ c_n\right\}$ be any numerical sequence. Does there exist a function $f\in \mathcal D(\mathbb {R})$ for which $f^{(n)}(0)=c_n$, $n=0,1,2,\ldots$?

Prove that any function of the form

is infinitely differentiable on $\mathbb {R}$ and that by an appropriate choice of the numbers $c_{n}$ one can arrange for all its derivatives at the point $x=0$ to take any prescribed values in advance (Borel's theorem).

Prove that to every sequence $\left\{ \alpha_{n}\right\}$ of complex numbers there corresponds a function $f \in C^{\infty }$ such that $\left(D^{n} f\right)(0) = \alpha_{n}$ for $n = 0,1,2, \ldots$ Suggestion: If $\varphi$ is as in Exercise 11, if $\beta_{n} = \alpha_{n} / n!$, if $g_{n}(x) = \beta_{n} x^{n} \varphi (x)$, and if

then $\left\| D^{k} f_{n}\right\|_{\infty } < 2^{-n}$ for $k = 0, \ldots , n-1$, provided that $\lambda_{n}$ is large enough. Take $f = \Sigma f_{n}$.