A numerical method is proven to be 1st-order accurate for $u' = F(u)$ and $u' = G(u)$, $t \in (0,T]$. Let $u_{F}(t)$ and $u_{G}(t)$ be the exact solutions of the two equations respectively. Let $v_{F}(t), \:\: v_{G}(t)$ be the numerical solutions respectively. Let $|w_{F}(t)|, \:\: |w_{G}(t)|$ be the error respectively. 1st-order accurate means there are constants $K_{F},K_{G} > 0$ such that
$$ |w_{F}| \le K_{F} \triangle t, \:\:\:\: |w_{G}| \le K_{G} \triangle t, \:\:\:\: \text{as} \:\: \triangle t \rightarrow 0$$
for all $t \in [0, T]$.
Does this imply the ODE $u' = F(u) + G(u)$ is also 1st-order accurate?
If not, what are some proper or clever ways to analyze this?