User:Keith.U/Sandbox/Proof 1

Theorem
Let $e$ denote Euler's Number.

Then $e \in \R$.

Proof
This proof assumes the series definition of $\exp$.

That is, let:
 * $\displaystyle e = \sum_{k \mathop = 0}^{\infty} \dfrac{1}{k!}$

From Series of Power over Factorial Converges, with $x = 1$:
 * $\displaystyle \sum_{k \mathop = 0}^{\infty} \dfrac{1}{k!}$  converges

Hence the result.