For questions about finding upper or lower bounds for functions (discrete or continuous).

**Definitions**

Given a function $f$ with domain $D$ and a partially ordered set $(K, \le)$ as codomain, an element $y \in K$ is an *upper bound* of $f$ if $f(x) \le y$ for each $x \in D$. The upper bound is called *sharp* if equality holds for at least one value of $x \in D$.

Function $g$ defined on domain $D$ and having the same codomain $(K, \le)$ is an upper bound of $f$ if $f(x) \le g(x)$ for each $x \in D$.

Function $g$ is further said to be an upper bound of a set of functions if it is an upper bound of each function in that set.

The notion of lower bound for (sets of) functions is defined analogously, with $\ge$ replacing $\le$.

An upper bound is said to be a *tight upper bound*, a *least upper bound*, or a *supremum* if no smaller value is an upper bound.
Similarly, a lower bound is said to be a *tight lower bound*, a *greatest lower bound*, or an *infimum* if no greater value is a lower bound.

Source: Wikipedia

**Examples**

For a random variable $X$ and $a > 0$: $$\Bbb{P}(X \ge a) \le \dfrac{\Bbb{E}(X)}{a}$$

This inequality is called Markov's inequality and provides an upper bound for the probability that the value of $X$ exceeds $a$.

For any real $x \ge 0 $: $$1 - e^{-x} \le x$$ So $x$ is an upper bound for $1-e^{-x}$, and vise versa, $1-e^{-x}$ is a lower bound in $[0; +\infty)$.