Let $D \subseteq \mathbb{R}^d$ be compact and nonempty. Furthermore, let $W^{(n)},W \in \mathbb{R}^{m \times d}$ be matrices and $b^{(n)}, b \in \mathbb{R}^m$ be vectors, such that
$$ \max_{i,j} \lvert W-W^{(n)} \rvert \rightarrow 0 \tag{1}$$
and
$$ \max_{i,j} \lvert b-b^{(n)} \rvert \rightarrow 0. \tag{2}$$
Now let $\rho \in \mathcal{C}(\mathbb{R}, \mathbb{R})$ be continuous and denote with $\rho_m$ the componentwise application of $\rho$ to a vector in $\mathbb{R}^m$.
Then I want to show, that $$f_n : D \rightarrow \mathbb{R}^m, \quad f_n (x) := \rho_m (W^{(n)}x+b^{(n)})$$ converges locally uniformly to $$f : D \rightarrow \mathbb{R}^m, \quad f (x) := \rho_m (Wx+b)$$ on $D$. In the paper, where I found this result, it says that this is supposed to be "easy to show". However, I only feel this is easy if $\rho$ is at least assumed to be locally Lipschitz continuous (which it is not).
Kind regards and thanks for any help, Joker
Hint: As $D$ is compact, your assumptions (1) and (2) yield that there is some $K>0$ such that $W^{(n)}x+b^{(n)}\in[-K,K]^m$ for all $x\in D$ and $n\in\mathbb N$. Then use that $\rho$ is uniformly continuous on $[-K,K]$ and that the functions defined by $\phi_n(x):=W^{(n)}x+b^{(n)}$ converge to $\phi(x):=Wx+b$ uniformly on $D$.