Real Part as Mapping is Surjection

Theorem
Let $f: \C \to \R$ be the projection from the complex numbers to the real numbers defined as:
 * $\forall z \in \C: \map f z = \map \Re z$

where $\map \Re z$ denotes the real part of $z$.

Then $f$ is a surjection.

Proof
Let $x \in \R$ be a real number.

Let $y \in \R$ be an arbitrary real number.

Let $z \in \C$ be the complex number defined as:
 * $z = x + i y$

Then we have:
 * $\map \Re z = x$

That is:
 * $\exists z \in \C: \map f z = x$

The result follows by definition of surjection.

Also see

 * Imaginary Part as Mapping is Surjection