---
tag: 0M1MX28VXK
kind: statement
role: theorem
name: Bernoulli's inequality
msc: [26D05]
uses: [0ASJVEFA9J]
sources:
  - url: https://en.wikipedia.org/wiki/Bernoulli%27s_inequality
    locator: Integer exponent
same_as:
  wikidata: Q728662
  wikipedia: 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|0ASJVEFA9J]] on $n$ multiplies $(1 + x)^n \ge 1 + n x$ by $1 + x \ge 0$ and uses $n x^2 \ge 0$.

## Explanations

### L1 human

A small growth rate $x$, applied $n$ times, gives at least $n$ times that growth.
For example, $1.1^{10}$ is at least $1 + 10 \cdot 0.1 = 2$; its true value is about $2.59$.
The difference comes from the growth on the growth, which the right side ignores.