# Exponential of Real Number is Strictly Positive/Proof 2

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## 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:

\(\ds \exp x\) | \(=\) | \(\ds \lim_{n \mathop \to \infty} \map {f_n} x\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds \lim_{n \mathop \to \infty} \map {f_{n + N} } x\) | Tail of Convergent Sequence | |||||||||||

\(\ds \) | \(\ge\) | \(\ds \map {f_{n + N} } x\) | Exponential Sequence is Eventually Increasing and Limit of Bounded Convergent Sequence is Bounded | |||||||||||

\(\ds \) | \(>\) | \(\ds 0\) | Corollary to Exponential Sequence is Eventually Increasing |

$\blacksquare$