The impulse response of an ideal low-pass filter can be determined by setting $H(\omega)=1$ in the Fourier-representation $$h(n) = \frac{1}{2\pi}\int_{-\omega_c}^{\omega_c} H(\omega)e^{j\omega n}d\omega$$
The solution will be a function of form $\sin(n)/n$. Now, the motivation behind the substitution above is the desire to amplify each frequency component before $\omega_c$ by an equal amount. Hence the term ideal response. Also, the fact that for some $\omega_0$ there is a corresponding amplitude $\vert H(\omega_0)\vert$ is not ( as far as I know ) interesting while performing spectral analysis. What matters are the amplitude relations between some $\omega_0$ and $\omega_1$, the frequency components of a signal perhaps.
Therefore, the substitution $H(\omega) = C$, for any constant $C$ should be equally valid right?