# Basic Inequalities in Analysis

Theorem 1. (Bernoulli’s inequality, lower bound for exponentiations of $$1 + x$$) $(1 + x)^r \geq 1 + rx$ for all $$r \in \mathbb{N}$$ and $$x \geq -1$$, $$x \in \mathbb{R}$$.

Theorem 2. (Generalization of Bernoulli’s inequality) $(1 + x)^r \geq 1 + rx$ for all $$r \in (-\infty, 0] \cup [1, +\infty)$$ and $$x \geq -1$$, $$x \in \mathbb{R}$$. $(1 + x)^r \leq 1 + rx$ for all $$r \in [0, 1]$$ and $$x \geq -1$$, $$x \in \mathbb{R}$$.

Theorem 3. $\left(1 + \frac{1}{x}\right)^x < e < \left(1 + \frac{1}{x}\right)^{x+1}$ for all $$x \in \mathbb{R}^+$$.

Theorem 4. (Upper bound for exponentiations of $$1 + x$$) $(1 + x)^r \leq e^{rx}$ for all $$r \in \mathbb{N}$$ and $$x \geq 0$$, $$x \in \mathbb{R}$$.

Theorem 5. For all $$x \in \mathbb{R}$$ it holds that $$e^x \geq x + 1$$.

Theorem 6. For all $$x \geq 1$$ it holds that $$(1 - \frac{1}{x})^x \leq e^{-1}$$.

Theorem 7. For all $$0 \leq x \leq 1$$ it holds that $e^{-x} \leq 1 - (1 - \frac{1}{e}) \cdot x \leq 1 - \frac{x}{2}$