Exponential of Real Number is Strictly Positive/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: