The Bernoulli inequality states
 |
(1)
|
where 
is a real number and 
an integer.
This inequality can be proven by taking a Maclaurin series of 
,
 |
(2)
|
Since the series terminates after a finite number of terms for integral 
, the Bernoulli inequality for 
is obtained by truncating after the first-order term.
When 
,
slightly more finesse is needed. In this case, let 
so that 
, and take
 |
(3)
|
Since each power of 
multiplies by a number 
and since the absolute value
of the coefficient of each subsequent term is smaller
than the last, it follows that the sum of the third order and subsequent terms is
a positive number. Therefore,
 |
(4)
|
or
 |
(5)
|
completing the proof of the inequality over all ranges of parameters.
For 
,
the following generalizations of Bernoulli inequality are valid for real exponents:
 |
(6)
|
and
 |
(7)
|
(Mitrinović 1970).
