Exponential series/Real/Elementary properties/Fact/Proof

Proof

(1) follows directly from the definition.
(2) follows from

${}\exp x\cdot \exp {\left(-x\right)}=\exp {\left(x-x\right)}=\exp 0=1\,$

using fact.
(3) follows for ${}n\in \mathbb {N}$ from fact by induction, and from that it follows with the help of (2) also for negative ${}n$.
(4). Nonnegativity follows from

${}\exp x=\exp {\left({\frac {x}{2}}+{\frac {x}{2}}\right)}=\exp {\frac {x}{2}}\cdot \exp {\frac {x}{2}}={\left(\exp {\frac {x}{2}}\right)}^{2}\geq 0\,.$

(5). For real ${}x$ we have ${}\exp x\cdot \exp {\left(-x\right)}=1$, so that because of (4), one factor must be ${}\geq 1$ and the other factor must be ${}\leq 1$. For ${}x>0$, we have

${}\exp x=\sum _{n=0}^{\infty }{\frac {1}{n!}}x^{n}=1+x+{\frac {1}{2}}x^{2}+\ldots >1\,$

as only positive numbers are added.
(6). For real ${}y>x$, we have ${}y-x>0$, and therefore, because of (5) ${}\exp {\left(y-x\right)}>1$, hence

${}\exp y=\exp {\left(y-x+x\right)}=\exp {\left(y-x\right)}\cdot \exp x>\exp x\,.$

To fact