Showing that if the $n$th derivative of a function is bounded then it is real analytic I reproduce from my lecture notes:

Suppose $f$ is $C^\infty$ on $[a,b]$ with $$\left|f^{(n)}(x)\right|\leqslant M~~\text{for all}~~x\in(a,b).$$ Then $f$ is real analytic in $[a,b]$.
Proof. Fix $x_0\in[a,b]$.  Then for each $x$ in $[a,b]$ the error after truncating the series at the $n$th term is given, for some $c\in[a,b]$, by $$\left|\frac{f^{(n+1)}(c)}{(n+1)!}(x-x_0)^{n+1}\right|\leqslant \frac{M}{(n+1)!}\left|b-a\right|^{n+1}.\tag{$*$}$$This tends to zero as $n\to\infty$ and so $f$ is real analytic in $[a,b]$.

Question: how did my lecturer deduce the inequality $(*)$ from $\left|f^{(n)}(x)\right|\leqslant M$? I am searching for different versions of this proof but they all somehow fail to explain this step, assuming that is is obvious. Any (fairly elementary) explanation is welcomed.
Thanks.
 A: As indicated in the comments, it would perhaps be better to rephrase the hypothesis as follows: Suppose $f$ is $C^\infty$ on $[a,b]$ and there exists a real number $M$ such that $|f^{(n)}(x)| \leq M$ for all $x \in [a,b]$ and all $n \in {\bf Z}_{>0}$. The assertion is then an immediate consequence of this hypothesis.
As a helpful example to keep in mind, consider the function $f(x)$ on ${\bf R}$ defined by $f(x)=e^{-1/x^2}$ for $x \neq 0$ and $f(0)=0$. You can check that this function is $C^\infty$ (in fact, it is the essential building block needed for partitions of unity on $C^\infty$ manifolds) with $f^{(n)}(0)=0$ for all $n \in {\bf Z}_{\geq 0}$. Therefore it is not analytic since its Maclaurin series has all coefficients equal to $0$. 
In fact, away from zero the derivatives of $f$ are of the form $f^{(n)}(x)=P_n(x) e^{-1/x^2}$ where $P_n(x)$ are polynomials of $x^{-1}$ determined by $P_0=1$ and $P_{n+1}=2x^{-3} P_n+P_n'$, hence with leading term $2^n x^{-3n}$. In particular for any fixed non-zero $x$ near to $0$, the sequence $P_n(x) e^{-1/x^2}$ goes to infinity with $n$. This shows directly that there is no bound as in your hypothesis for this function (as there cannot be if the assertion is to be believed).
