## Bernoulli Inequality

 (1)

where , . 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.