ProofWiki:Sandbox

Theorem
The complex exponential function is its own derivative.

That is:
 * $D_z \left({\exp z}\right) = \exp z$

Proof from Sequence Definition
Take the definition of $\exp$ to be the limit of the sequence $\left \langle{E_n}\right \rangle$ defined by:
 * $\displaystyle E_n \left( {z} \right) = \lim_{n \to \infty} \left({1 + \dfrac z n}\right)^n$

Then $\left \langle{E_n}\right \rangle$ is uniformly convergent on compact subsets of $\C$.

Further, $\C$ is an open,  connected subset of $\C$.

So the hypotheses of  Derivative of Sequence of Holomorphic Functions are satisfied.

Hence:

Hence the result.