Characteristic of Subfield of Complex Numbers is Zero

Theorem
The characteristic of any subfield of the field of complex numbers is $0$.

Proof
to the contrary.

Let $K$ be a subfield of $\C$ such that $\Char K = n$ where $n \in \N, n > 0$.

Then from Characteristic times Ring Element is Ring Zero:
 * $\forall a \in K: n \cdot a = 0$

But as $K$ is a subfield of $\C$ it follows that $K \subseteq \C$ which means:
 * $\exists a \in \C: n \cdot a = 0$

Thus, by definition of characteristic:
 * $0 < \Char \C \le n$

But $\C$ is infinite and so $\Char \C = 0$.

From that contradiction follows the result.