Real numbers/Bernoulli's inequality/Fact/Proof

Proof

We do induction over ${}n$.  Suppose that the statement is already known for ${}n$. Then

${}{\begin{aligned}(1+x)^{n+1}&=(1+x)^{n}(1+x)\\&\geq (1+nx)(1+x)\\&=1+(n+1)x+nx^{2}\\&\geq 1+(n+1)x,\end{aligned}}$

since squares in an ordered field are nonnegative.

To fact