(1) follows directly from the definition.
(2) follows from
-
![{\displaystyle {}\exp x\cdot \exp {\left(-x\right)}=\exp {\left(x-x\right)}=\exp 0=1\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a6089865636d44d0107a860e64c63b726739658b)
using
fact.
(3) follows for
from
fact
by induction, and from that it follows with the help of (2) also for negative
.
(4). Nonnegativity follows from
-
![{\displaystyle {}\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\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b5a86daefdd5e9edf25670eaaee6ab586f22c896)
(5). For real
we have
,
so that because of (4), one factor must be
and the other factor must be
. For
,
we have
-
![{\displaystyle {}\exp x=\sum _{n=0}^{\infty }{\frac {1}{n!}}x^{n}=1+x+{\frac {1}{2}}x^{2}+\ldots >1\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f622865b39785e6000b3a2fdc37c8fad96c18fcb)
as only positive numbers are added.
(6). For real
,
we have
,
and therefore, because of (5)
,
hence
-
![{\displaystyle {}\exp y=\exp {\left(y-x+x\right)}=\exp {\left(y-x\right)}\cdot \exp x>\exp x\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0fb483636ce2ebbfdf3dfee31f2e545c20639c4f)