# 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 | $\quad$ |

$\blacksquare$