Uniform continuity of characteristic function of a tight family of measure

Question

I am missing a step in the proof of Theorem 15.22 of Probabiliy Theory by A. Klenke (3rd version).

The theorem states that, given a tight family of probability measure on $\mathbb{R}$, the family of the characteristic functions is uniformly continuous.

I don’t get this first inequality in the proof:

$1-Re(\phi_{\mu}(u)) \\\leq \int |1-\exp(ixu)|\mu(dx)$

Where $|u|$ is “small”.

Any suggestion?

Edit

Solution:

$1-Re(\phi_{\mu}(u)) \\ =Re(1-\phi_{\mu}(u))\\ \leq |1-\phi_{\mu}(u)|\\ \leq \int |1-\exp(ixu)|\mu(dx)$