Let $f:\mathbb{R}\longrightarrow \mathbb{R}$ is of class ${C}^{\infty}$. Suppose that for every $a, b \in\mathbb{R}\ (a<b)$ there exists $K>0$ (possibly depending on $a$ and $b$) such that $|f^{(n)}(x)|\leq K$ for every $x\in [a, b]$ and every $n\in \mathbb{N}$.
Show that $f$ is globally analytic.
My attempt: I tried to use Taylor series, but I got nowhere!
Any comment?
Hint: Try using the bound to show that the the error term for the Taylor series goes to zero over all intervals.