Bernoulli's Inequality/Corollary/Proof 1

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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:

$\left({1 - x}\right)^n \ge 1 - n x$


Proof

Let $0 < x < 1$.

Let $y = -x$.

Then $y > -1$ and by Bernoulli's Inequality:

$\left({1 + y}\right)^n \ge 1 + n y$

Thus:

$\left({1 + \left({-x}\right)}\right)^n \ge 1 + n \left({-x}\right)$

Hence the result.

$\blacksquare$