# Exponential series/Real/Elementary properties/Fact/Proof

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Proof

(1) follows directly from the definition.

(2) follows from

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

(5). For real we have , so that because of (4), one factor must be and the other factor must be . For , we have

as only positive numbers are added.

(6). For real
,
we have
,
and therefore, because of (5)
,
hence