Sums of equidifferentiable functions are themselves equidifferentiable?

Question

Suppose the sequence of vector valued functions $\{ {\bf f}_n \}$ are equidifferentiable at ${\bf x}_0$. In other words:

$$\lim_{{\bf h} \to {\bf 0}} \max_n \frac{\left\Vert {\bf f}_n({\bf x}_0+{\bf h}) - {\bf f}_n({\bf x}_0) - \triangledown {\bf f}_n({\bf x}_0) {\bf h} \right\Vert}{\left\Vert {\bf h} \right\Vert} = 0$$

Let

$${\bf g}_n(x) = \sum_{k=1}^{n} w_k {\bf f}_k(x)$$

Question: Which restrictions on the $w_k$ are necessarily to ensure that $\{{\bf g}_n(x) \}$ are also equidifferentiable at ${\bf x}_0$?

Answer 1

Define

$$r_n({ \bf h}) = \frac{\left\Vert {\bf f}_n({\bf x}_0+{\bf h}) - {\bf f}_n({\bf x}_0) - \triangledown {\bf f}_n({\bf x}_0) {\bf h} \right\Vert}{\left\Vert {\bf h} \right\Vert} $$

A sufficient condition for equidifferentiability of the weighted sum is that

$$\max_n |w_k r_n({\bf h})| = o(1) $$

A sufficient condition for it is that $|w_k|$ be bounded and $\{ {\bf f}_n\}$ be equidifferentiable.