Derivative of Exponential Function/Complex

Theorem
The complex exponential function is its own derivative.

That is:
 * $\map {D_z} {\exp z} = \exp z$

Proof from Sequence Definition
Take the definition of $\exp$ to be the limit of the sequence $\sequence {E_n}$ defined by:
 * $\ds \map {E_n} z = \paren {1 + \dfrac z n}^n$

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.