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:

$\displaystyle \exp x = \lim_{n \mathop \to \infty} \map {f_n} x$

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:

\(\displaystyle \exp x\)

\(=\)

\(\displaystyle \lim_{n \mathop \to \infty} \map {f_n} x\)

$\quad$

$\quad$

\(\displaystyle \)

\(=\)

\(\displaystyle \lim_{n \mathop \to \infty} \map {f_{n + N} } x\)

$\quad$ Tail of Convergent Sequence

$\quad$

\(\displaystyle \)

\(\ge\)

\(\displaystyle \map {f_{n + N} } x\)

$\quad$ Exponential Sequence is Eventually Increasing and Limit of Bounded Convergent Sequence is Bounded

$\quad$

\(\displaystyle \)

\(>\)

\(\displaystyle 0\)

$\quad$ Corollary to Exponential Sequence is Eventually Increasing

$\quad$

$\blacksquare$