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:

\(\displaystyle \exp x\)

\(=\)

\(\displaystyle \lim_{n \to \infty} f_n \left({x}\right)\)

$\quad$

$\quad$

\(\displaystyle \)

\(=\)

\(\displaystyle \lim_{n \to \infty} f_{n + N} \left({x}\right)\)

$\quad$ Tail of Convergent Sequence

$\quad$

\(\displaystyle \)

\(\ge\)

\(\displaystyle f_{n + N} \left({x}\right)\)

$\quad$ 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

$\quad$

$\blacksquare$