Proving image of nonconstant, entire function is dense in $\mathbb{C}$

Question

The problem at hand is:

let $f$ be a nonconstant and entire function.

1. Prove that for all $\omega \in \mathbb{C}$ and for all $\varepsilon > 0 $ there exists $z \in \mathbb{C}$ such that $|f(z) - w| < \varepsilon $

2. Deduce that $f(\mathbb{C})$ is dense in $\mathbb{C}$

3. Is $f$ necessarily onto?

We have not taken the Casorati-Weierstrass theorem yet but I figured this is a variation from it from what I've read on the net. I don't think I understand the approaches for proofs so far..

My attempt at 1 is:

$f$ is entire, therefore it is continuous.

Then for all $\varepsilon > 0$ there exists $\delta$ such that taking $z_1 \in \mathbb{C}$, $|z - z_1| < \delta$ implies $|f(z) - f(z_1)| < \varepsilon$

Therefore, letting $\omega = f(z_1)$ for any $z_1 \in \mathbb{C}$ we get the result that for any $\omega$ there exists a $z$ such that $|f(z) - w| < \varepsilon $

Is this approach okay? I did not use the fact that the function is nonconstant. How does that relate to density?