Let $u(t)$ be the numerical solution of a differential equation $y'(t)=f(t,y)$ for $t \in (0, T]$ with initial condition $y_{0}$. Let $|w(t)|=|u(t)-y(t)|$ be the error function.
I have proven that there is a constant $C$
$$|w(t)| \le C \triangle t, \:\:\: \triangle t \rightarrow 0$$
but the constant $C$ itself depends on the period of the domain, $T$. Is it a valid proof of 1st-order convergence if the constant $C$ depends on $T$?
Yes. Assuming that your inequality is valid for the entire interval, then the only requirement is that your constant is independent of your time step.