Real functions Classical inequalities Theorem

Bernoulli's inequality

Statement

Let $x$ be a real number with $x \ge -1$ , and let $n$ be a natural number. Then $(1 + x)^n \ge 1 + n x.$

Remarks

When $x \neq 0$ and $n \ge 2$ , the inequality is strict. For natural numbers $n$ , the inequality also holds when $x \ge -2$ . A proof by induction on $n$ multiplies $(1 + x)^n \ge 1 + n x$ by $1 + x \ge 0$ and uses $n x^2 \ge 0$ .

Reading path

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Mathematical induction concept
Bernoulli's inequality statement

Tag	`0M1MX28VXK`
URI	`m2://0M1MX28VXK`
Kind	statement, theorem
Writing	W1 stated
MSC	26D05
Status	This copy of the site has no build data. Run `m2 status 0M1MX28VXK`.
Uses	Mathematical induction
Same as	Wikidata Q728662 Wikipedia Bernoulli's inequality

Bernoulli's inequality

Statement

Remarks

Reading path

Further reading

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