On sequences of $C^{k,\delta}_{b}$ maps which converge in the uniform $C^{0}$-norm

We study maps $f\in C^{k,\delta}_{b}(U,Y)$ and give detailed estimates on $\Vert D^{k}f(x)\Vert,x\in U,$ in terms of $\Vert f\Vert$ and $\Vert f\Vert _{k,\delta}$. These estimates are used to prove a lemma by D. Henry for the case $k\geq 2$. Here $U\subset X$ is an open subset and $X$ and $Y$ are Banach spaces.