Application of Baire Category theorem

Question

Suppose that $f$ is infinitely differentiable on $[a,b]$ and suppose that for any $a ≤ x ≤ b$ the Taylor series of $f$ has positive radius of convergence at $x$. Use the Baire Category Theorem to show that $f$ must be analytic on a subinterval of $[a,b ]$.

Answer 1

This is taken from the argument given in the link in my comment above. See the link for details.

Hint:

For $k$ a positive integer, define $$ E_k=\bigl\{ \,y\in [a,b]\,\mid \, \sup_n |f^{(n)} (y)/ n!|^{1/n}< k\,\bigr\}. $$

For a given $y$, the quantity $\sup_n |f^{(n)} (y)/ n!|^{1/n}$ is finite, since the Taylor series about $y$ has positive radius of convergence. Use Baire to show some $E_k$ is not nowhere dense.

From the subinterval obtained via the appeal to Baire, argue that the Lagrange form of the Taylor remainder at a $y$ in this subinterval tends to $0$ for every $x$ sufficiently close to $y$.