Fix some $d\in\mathbb{N}$, $m\in \mathbb{N}_0$ and $0<a\leq 1$. Let $f_1,..., f_n$ be functions on $[0,1]^d$ with Hölder smoothness $m+a$. That is, they are $m$-times continuously differentiable and the $m$-th order partial derivatives are $a$-Hölder. For such functions consider the Hölder semi-norm $$||f||_H = \sum_{\gamma\, :\, |\gamma|_1 \leq m} ||\partial^\gamma f||_\infty + \sum_{\gamma\,:\,|\gamma|_1=m} \sup_{x,y\in[0,1]^d, x\neq y} \frac{|\partial^\gamma f(x)-\partial^\gamma f(y)|}{|x-y|_\infty^a},$$ where $\gamma\in\mathbb{N}_0^d$.
Suppose there is some $K>0$ such that $||f_i||_H\leq K$ for all $i$ and suppose that for each $i,j$ with $i\neq j$, the functions $f_i$ and $f_j$ have disjoint support.
I am interested in bounding the Hölder semi-norm of $\sum_{i=1}^n f_i$. A proof I'm reading seems to imply that we should be able to bound $||\sum_{i=1}^n f_i||_H\leq 2K$, but I haven't been able to prove this.
Any help would be appreciated!