Characteristic function and first moment of random variable

Question

Let $\phi_X$ denote the characteristic function of a random variable $X$ with finite first moment. I want to prove the following inequality

$$ |\phi_X(u) - 1| \leq |u| \times \mathbb{E}\left[\left|X\right|\right].$$

I was thinking of using the mean value theorem since $ |\phi_X(u) - 1| = |\phi_X(u) - \phi_X(0)|$ but I fail to link it to the first moment.

Thanks.