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 $$e^{-x} \leq 1 - (1 - \frac{1}{e}) \cdot x \leq 1 - \frac{x}{2}$$