How can I prove $left|frac{e^{it_p x_j}-1}{t_p}right| leq 2|x|$?

Question

Background:
I am trying to understand why, if $X$ is a random variable with $mathbb{E}big[|X|big]< +infty$ and $mu=mathbb{P}^{X}$, then $$frac{partial}{partial x_j}overline{mu}(u)=iint e^{i langle u,x rangle} x_j mu(dx).$$
My problem:
Suppose $t_p to 0$ and $x in mathbb{R}^n$. How can I prove that $$left|frac{e^{it_p x_j}-1}{t_p}right| leq 2|x|,$$ where $x_j$ is the $j$-th coordinate of $x$?

Kavi Rama Murthy · Accepted Answer

In fact you can get $|x|$ as a bound. Use the fact that $|e^{itheta} -1| leq |theta|$ for all real numbers $theta$.
[$int_0^{1} e^{ittheta} dt=frac  1 {i theta} (e^{itheta}-1)$ and LHS is bounded in modulus by $int_0^{1}dt=1$].

How can I prove $left|frac{e^{it_p x_j}-1}{t_p}right| leq 2|x|$?

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