$\sum |f_n|$ converges uniformly implies $\sum f_n$ converges normally Is the following proposition true?

Let $f_n \colon U\subset\mathbb{C}\longrightarrow \mathbb{C}$ be a sequence of continuous (or holomorphic) functions such that:
$\sum |f_n|$   converges uniformly  on every compact $K\subset U$
Then $\sum f_n$ converges normally on every compact $K\subset U$
i.e. $\sum ||f_n||_K < \infty$ where $||f_n||_K =\sup_{z\in K} |f_n(z)|$

Any help would be appreciated.
 A: I give a counterexample for   the continuous case only; the parenthetical "(or holomorphic)" should really be a separate question.
Let $U$ be the unit disk and $K$ be the closed disk of radius $1/2$. The series of constant functions $g_n = 1/n^2$ converges uniformly on $U$. Use a continuous partition of unity to write 
$$g_n = h_{n,1}+\dots + h_{n,n}$$
where $0\le h_{n,k}\le 1/n^2$ and $\max_K h_{n,k}=1/n^2$ for every $k$. One way to do this is to pick $n  $ distinct points $z_k\in K$, let $\delta$ be the smallest distance between them, and define
$$\begin{split}
h_{n,k}(z) &= \frac{1}{n^2}(1-2\delta^{-1}|z-z_k|)^+,\quad k=1,\dots,n-1  \\
 h_{n,n} &= g_n-\sum_{k=1}^{n-1} h_{n,k}\end{split}$$
The series $$h_{1,1}+h_{2,1}+h_{2,2}+h_{3,1}+h_{3,2}+h_{3,3}+h_{4,1}+\cdots$$
converges uniformly on $U$, since every partial sum is sandwiched between two partial sums of $\sum g_n$. On the other hand,  the sum of suprema of its terms on $K$ is 
$$\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{3^2}+\frac{1}{3^2}+\frac{1}{4^2}+\cdots$$ which diverges. 
