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:
 * $\ds \exp x = \lim_{n \mathop \to \infty} \map {f_n} x$

where $\map {f_n} x = \paren {1 + \dfrac x n}^n$

First, fix $x \in \R$.

Let $N = \ceiling {\size x}$, where $\ceiling {\, \cdot \,}$ denotes the ceiling function.

Then: