Bounded Set¶

Definition¶

A set $X\subseteq \mathbb{R}$ is said to be bounded if there exists a real number $M>0$ such that $X \subseteq [-M, M]\text{,}$ i.e., this set can be contained within a bounded interval. A set that is not bounded is referred to as an unbounded set.

Interpretation¶

A bounded set consists of elements that all lie within some finite distance of $0$: one number $M$ satisfies $|x|\le M$ for every $x\in X$$X$. The set does not reach arbitrarily far in either direction.

The set $X$ in the figure is made of an interval and three isolated points, yet all of it lies inside the shaded $[-M,M]$. A bounded set need not be an interval; it only has to fit inside one.

Bounded means bounded above and bounded below: if $X\subseteq[a,b]$$[a,b]$, then $X\subseteq[-M,M]$ with $M=\max\{|a|,|b|,1\}$$\max \{|a|,|b|,1\}$, so the centre $0$ in the definition is only a convenience. A nonempty bounded set has a supremum and an infimum, but not necessarily a maximum or a minimum: $(0,1)$ has neither.

A common slip is to confuse bounded with finite. Every finite set is bounded, but $[0,1]$ and $\{\frac{1}{n}\colon n\ge1\}$ are infinite and still bounded, while $\mathbb{N}$ is unbounded although each of its elements is an ordinary finite number.

Key Results¶

• Every bounded set is contained in a bounded interval.
• Every bounded interval is a bounded set.
• Every finite set is bounded.

Examples¶

• The set $\{x\in\mathbb{Q}\colon |x|\le 3\}$ is bounded.
• $\mathbb{N}$ is not bounded.

Prerequisites¶

• Endpoint and Bounded Interval

Related Concepts¶

• Compact Set
• Bounded Function
• Bounded Set in a Metric Space
• Infimum
• Lower Bound
• Supremum
• Upper Bound

Sources¶

• Math I: Calculus · Lecture 4

Symbols: element of, natural numbers, rational numbers, real numbers, subset

Last updated: 20 October 2026 03:02