Inequality for complex exponential function

Question

It is easy to proof that for $x \in \mathbb{R}$ $$ \mid e^{ix} \mid =1 $$ But my lecture notes also contains this: For $x,u \in \mathbb{R}^d$ $$ \mid e^{i<x,u>}\mid\leq 1 $$ How do I show that?

Answer 1

$\langle x, u \rangle$ is a real number. Hence $|e^{i\langle x, u \rangle}|=1$, in fact.