User:Keith.U/Sandbox/Proof 5

Theorem
Let $e$ denote Euler's Number.

Then $e \in \R$.

Proof
This proof assumes the Base of Exponential with Derivative $1$ at $0$ definition of $e$.

That is, suppose:
 * $\displaystyle \lim_{h \to 0} \frac{ e^{h} - 1 }{ h } = 1$

and that such a number is Unique.

From Natural Logarithm as Derivative of Exponential at Zero:
 * $\displaystyle \lim_{h \to 0} \frac{ e^{h} - 1 }{ h } = \ln e$

From Logarithm of e is 1, $\ln e = 1$.

Hence the result.