User:Keith.U/Sandbox/Proof 2

Theorem
Let $x$ be a real number.

Let $\exp$ denote the (real) Exponential Function.

Then:
 * $\forall x \in \R : \exp x > 0$

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

That is, let:
 * $\displaystyle \exp x = \lim_{n \to \infty} f_n \left({ x }\right)$

where $f_n \left({ x }\right) = \left({ 1 + \dfrac{x}{n} }\right)^{n}$

First, fix $x \in \R$.

Let $N = \left\lceil{ \left\vert{ x }\right\vert }\right\rceil$, where $\left\lceil{ \cdot }\right\rceil$ denotes the ceiling function.

Then: