Lower Bound¶

Definition¶

Let $E$ be a subset of $\mathbb{R}\text{,}$ and let $M$ be a real number. We say that $M$ is a lower bound for $E$ if we have $x \geq M$ for every element $x$ in $E\text{.}$

Interpretation¶

A lower bound of a set is a value that does not exceed any element of the set: it marks a level that the set never goes below. Lower bounds describe constraints from below without requiring attainability.

A number line with a set E made of the point -1.1 and the interval from -0.5 (open) to 2, with its lower bounds shaded up to -1.1; the greatest lower bound -1.1 belongs to E.

Here $E$ consists of the point $-1.1$ and the numbers in $(-0.5,2]$ . The marked $m_1$ , $m_2$ and $m_3$ are lower bounds, and so is every $m\le-1.1$ $- 1.1$ ; the greatest one is $\inf E=-1.1=\min E$ $min E$ . The number $-1$ is not a lower bound, because the single element $-1.1$ lies below it.

The set $E$ specifies which elements must be compared. The inequality $M\le x$ must hold for every $x\in E$ $E$ ; $M$ need not be an element of $E$ . Lower bounds are never unique: if $M$ is one, so is every smaller number. The useful question is therefore which lower bound is the largest, and that is the infimum.

A common slip is to negate the definition incorrectly. $M$ fails to be a lower bound as soon as one element $x\in E$ satisfies $x<M$ $M$ . So $\mathbb{Z}$ has no lower bound, every nonempty finite set has its smallest element as one, and for the empty set every real number is a lower bound, since there is nothing to compare.

Key Results¶

If $M$ is a lower bound for $E$ , then so is every $M'\le M\text{;}$ hence lower bounds are never unique.
Lower bounds need not belong to the set.

Examples¶

Both sets $(0,1)$ and $\{0,1\}$ share the same lower bound $0\text{.}$
Any number less than $0$ is also a lower bound of $(0,1)\text{.}$

Prerequisites¶

Sources¶

Math I: Calculus · Lecture 3

Aliases: minorant Symbols: element of, integers, real numbers

Last updated: 20 October 2026 03:02