Quasi-analytic classes (Denjoy-Carleman)

Let $f \in C^{\infty }(\left[a, b\right]), f \not\equiv 0$. Prove that if the function $f$ vanishes on some nonempty interval, then

Does there exist a point $x$ such that $\sup_{n}\left|f^{(n)}(x)\right|=+\infty$?

Prove that $C\left\{ n!\right\} = C\left\{ n^{n}\right\}$.

Prove that there are quasi-analytic classes which are larger than $C\left\{ n!\right\}$.

Put $\lambda_{n} = M_{n-1} / M_{n}$, as in the proof of Theorem 19.11. Pick $g_{0} \in C_{c}\left(\mathbb {R}^{1}\right)$, and define

Prove directly (without using Fourier transforms or holomorphic functions) that $g = \lim g_{n}$ is a function which demonstrates that (e) implies (a) in Theorem 19.11. (You may choose any $g_{0}$ that is convenient).

Suppose $C\left\{ M_{n}\right\}$ is quasi-analytic, $f \in C\left\{ M_{n}\right\}$, and $f(x) = 0$ for infinitely many $x \in [0,1]$. What follows?