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...
$$|\mathop{\rm Re}(z)-\mathop{\rm Re}(z_0)| = |x - x_0| = |x| - |x_0| = x - x_0$$

We have 
$|z_0-z|^2=(\Re(z_0-z))^2+(\Im(z_0-z))^2\geq (\Re(z_0-z))^2$ so $|z_0-z|\geq |z_0-z|$. Now, check that $\delta=\varepsilon$ in the definition of the limit works.
