Is this proof correct?:
Proof:
Suppose that $X$ is a random variable on a probability space $\{\Omega, \mathcal{F}, \mathbb{P}\}$. Suppose $x \in \mathbb{R}$ and $x \geq 0$. Then $\{|X| \leq x\} = \{X \leq x \} \cap \{X < -x \}^c$. Since $X$ is a random variable $\{X \leq x \} \in \mathcal{F}$ and $\{X < -x \}^c \in \mathcal{F}$. Since $\mathcal{F}$ is a $\sigma$-field, the intersection $\{|X| \leq x\} = \{X \leq x \} \cap \{X < -x \}^c$ also belongs to $\mathcal{F}$. Thus $|X|$ is a random variable. $\square$
Best Answer
Your proof is correct.
You also could have argued with this more general statement: If $X$ is a random variable and $f$ is continuous, then $f(X)$ is a random variable [assuming you choose the Borel-sigma-algebra on the domain and the range of $f$].