This is a follow up on the discussion Convergence of a sequence of real convex analytic functions
Let $(f_n)_{n \in \mathbb N}$ be a sequence of convex analytic functions, with bounded derivatives (of all orders: that it exists a constant K independent of $n$ such that $f’(x)<K$, $f’’(x)<K$, etc...) on $\mathbb R$.
Suppose that $f_n(x)\to f(x)$ as $n \to \infty$ for all $x \in \mathbb R$ ( or in \mathbb R^+) (pointwise convergence)
Is it true that $f(x)$ is analytic?
NB: Clearly, this is not case without the bounded derivatives constraint (see this discussion).