**Bernoulli's inequality** in

real analysis states that

- (1 +
*x*)^{n} ≥ 1 + *nx*

for every

integer *n* ≥ 0 and every

real number *x* ≥ -1. The strict version of the inequality reads

- (1 +
*x*)^{n} > 1 + *nx*

for every integer

*n* ≥ 2 and every real number

*x* ≥ -1 with

*x* ≠ 0.

The inequality is often used as the crucial step in the proof of other inequalities. It can be proven using mathematical induction.