Bounded
Elementary
A set of numbers is bounded if it does not go on forever in either direction: there is some number that every number in the set is smaller than in size.
High school (sets of real numbers)
A set $S$ of real numbers is bounded if there is a number $M$ such that $\abs{x} \leq M$ for every $x \in S$.
Undergraduate (metric spaces)
A subset of a metric space is bounded if it is contained in some ball. A sequence or function is bounded if its set of values is bounded.
Examples
Examples
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$[0, 1]$
High school
$[0, 1]$ is compact because it is closed and bounded.
-
$(0, 1)$
High school
$(0, 1)$ is bounded.
- $\mathbb{Q} \cap [0, 1]$ High school
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$\setdef{1, 2, 3}$
High school
$\setdef{1, 2, 3}$ is a finite set, so it is bounded and compact, as is every finite subset of $\reals$.
Non-examples
Things that may be mistaken for examples.
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The sequence $1, 2, 3, \dots$
Elementary
The sequence $1, 2, 3, \dots$ is not bounded.
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$\nats$
High school
$\nats$ is not bounded, because for every $N > 0$, we can pick $n \in \nats$ such that $n > N$.
Distinctions involving this term
- Types of infinity: Infinity appears in several different senses: as the size of a set, as unboundedness, and as the limit of a divergent sequence. An infinite set has infinitely many elements, but it can be bounded. Infinite sets come in different sizes: countable and uncountable. A divergent sequence does not converge, but it doesn’t necessarily go to infinity.