Bernoulli's Inequality/Corollary/Proof 1
Jump to navigation
Jump to search
Theorem
Let $x \in \R$ be a real number such that $0 < x < 1$.
Let $n \in \Z_{\ge 0}$ be a positive integer.
Then:
- $\paren {1 - x}^n \ge 1 - n x$
Proof
Let $0 < x < 1$.
Let $y = -x$.
Then $y > -1$ and by Bernoulli's Inequality:
- $\paren {1 + y}^n \ge 1 + n y$
Thus:
- $\paren {1 + \paren {-x} }^n \ge 1 + n \paren {-x}$
Hence the result.
$\blacksquare$