# Exponential of x not less than 1+x

## Theorem

$e^x \ge 1 + x$

for all $x \in \R$.

## Proof

For $x > - 1$:

 $\displaystyle e^x$ $=$ $\displaystyle \lim_{n \mathop \to \infty} \left({1 + \frac x n}\right)^n$ Definition of Real Exponential Function $\displaystyle$ $\ge$ $\displaystyle \lim_{n \mathop \to \infty} \left({1 + x}\right)$ Bernoulli's Inequality $\displaystyle$ $=$ $\displaystyle 1 + x$

as required.

For $x \le -1$, the inequality follows from the fact that $e^x$ is positive for all $x$.

$\blacksquare$