# Proof of a CTFT property

How do I prove that the absolute value of CTFT of a positive continuous time signal is less than or equal to its value at central frequency(0)?

If $f(t)$ is a real non-negative function with Fourier transform
$$F(\omega)=\int_{-\infty}^{\infty}f(t)e^{-j\omega t}dt\tag{1}$$
$$|F(\omega)|=\left|\int_{-\infty}^{\infty}f(t)e^{-j\omega t}dt\right|\le\int_{-\infty}^{\infty}|f(t)|dt=\int_{-\infty}^{\infty}f(t)dt=F(0)$$
since $f(t)=|f(t)|$.