#### Answer

The statement is true. See below for proof.

#### Work Step by Step

We are given that $z=a+bi$:
$z-\overline{z}=(a+bi)-\overline{(a+bi)}=a+bi-(a-bi)=a+bi-a+bi=2bi+a-a=2bi$
We know that $b$ is a real number, so $2bi$ is a pure imaginary number.