Proving that the "real part" function is continuous I want to use the definition of a limit,
$|f(z) - w_0| < \varepsilon$ whenever $0 < |z - z_0| < \delta$
to prove
$$\lim_{z \to z_0} \mathop{\rm Re}(z) = \mathop{\rm Re}(z_0)$$
By intuition this is obvious but I dont know how to show it using the defn. of a limit. This is question 1(a) from the book Complex Variables and Applications.
Here's the basic manipulation I have made going by an example in the book, I dont know where to go from here...
$$|\mathop{\rm Re}(z)-\mathop{\rm Re}(z_0)| = |x - x_0| = |x| - |x_0| = x - x_0$$

If you have a complex number $z = a + ib$ where $a$ and $b$ are real, then
$$|z| = \sqrt{a^2 + b^2} \ge \sqrt{a^2} = |a|.$$
From this we see that for any complex number $z$,
$$|z| \ge |\Re(z)| \ge \Re(z).$$
Now apply this to $z - z_0$ and your limit follows right away.
