Exponential of Real Number is Strictly Positive/Proof 5

From ProofWiki
Jump to navigation Jump to search

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 definition of $\exp$ as the solution to an initial value problem.

That is, suppose $\exp$ satisfies:

$ (1): \quad D_x \exp x = \exp x$
$ (2): \quad \map \exp 0 = 1$

on $\R$.


Lemma

$\forall x \in \R: \exp x \ne 0$

$\Box$


Aiming for a contradiction, suppose that $\exists \alpha \in \R: \exp \alpha < 0$.


Then $0 \in \openint {\exp \alpha} 1$.

From Intermediate Value Theorem:

$\exists \zeta \in \openint \alpha 0: \map f \zeta = 0$


This contradicts the lemma.

$\blacksquare$


Sources