Infimum

From Upper Edges to Lower Edges

The previous tutorial introduced the supremum as the least upper bound of a set. The corresponding idea at the lower edge is the infimum: it is the greatest of all the set’s lower bounds. Like a supremum, an infimum need not belong to the set. It describes the best possible bound even when no element attains it.

Recall from “Lower Bounds” that \(L\) is a lower bound for \(S\subseteq\mathbb{R}\) when \(L\leq x\) for every \(x\in S\). To identify the infimum, we compare all of a set’s lower bounds and select the largest one.

The definition requires two things: \(\beta\) must be a lower bound, and it must be at least as large as every other lower bound. A minimum has the additional requirement that it belong to the set. Thus an infimum is a minimum exactly when it is attained in the set.

Existence of Infima

The Completeness Principle from “Supremum” states that every nonempty set of real numbers bounded above has a supremum. Applying that principle to the negatives of the elements of a set gives the corresponding existence result for infima.

Proof. Let \(S\subseteq\mathbb{R}\) be nonempty and bounded below, and define \(-S=\{-s:s\in S\}\). Since \(S\) has a lower bound \(L\), we have \(L\leq s\) for every \(s\in S\). Order reversal under negation gives \(-s\leq-L\), so \(-L\) is an upper bound for \(-S\). The set \(-S\) is nonempty, and the Completeness Principle therefore gives \(\gamma=\sup(-S)\).

Set \(\beta=-\gamma\). For any \(s\in S\), \(-s\in -S\), and \(\gamma\) is an upper bound for \(-S\). Hence \(-s\leq\gamma\). Negating this inequality gives \(\beta=-\gamma\leq s\). Thus \(\beta\) is a lower bound for \(S\).

Now let \(L'\) be any lower bound for \(S\). Then \(L'\leq s\) for every \(s\in S\), so \(-s\leq-L'\) for every \(s\in S\). This means \(-L'\) is an upper bound for \(-S\). Since \(\gamma=\sup(-S)\), we have \(\gamma\leq-L'\), and order reversal gives \(L'\leq-\gamma=\beta\). Therefore \(\beta\) is at least as large as every lower bound for \(S\). It is the infimum of \(S\), as required. \(\square\)

Both hypotheses in the theorem matter. A nonempty set unbounded below has no lower bounds and therefore cannot have an infimum as defined here. The empty set also has no infimum in \(\mathbb{R}\): every real number is a lower bound for it, and there is no greatest real number.

An Approximation Test for the Infimum

The definition compares the infimum with every lower bound. An equivalent test says that elements of the set can be found arbitrarily close to the infimum from above. If a proposed lower bound could be raised even slightly and still remain a lower bound, it would not be the greatest one.

Proof. First suppose \(\beta=\inf S\), and let \(\varepsilon>0\). If there were no \(s\in S\) with \(s<\beta+\varepsilon\), then every \(s\in S\) would satisfy \(\beta+\varepsilon\leq s\). That would make \(\beta+\varepsilon\) a lower bound for \(S\). But \(\beta+\varepsilon>\beta\), contradicting that \(\beta\) is the greatest lower bound. Thus some \(s\in S\) satisfies \(s<\beta+\varepsilon\).

Conversely, suppose that for every \(\varepsilon>0\) there is an \(s\in S\) such that \(s<\beta+\varepsilon\), and recall that \(\beta\) is a lower bound. Let \(L\) be any lower bound for \(S\). If \(L>\beta\), set \(\varepsilon=L-\beta\), which is positive. The assumed condition gives some \(s\in S\) with \(s<\beta+\varepsilon=L\). But \(L\) is a lower bound, so \(L\leq s\), a contradiction. Therefore \(L\leq\beta\) for every lower bound \(L\), proving that \(\beta=\inf S\). \(\square\)

Infima of Unions

The infimum of a union of two sets can be found by comparing the infima of the individual sets. The smaller of those two values is a lower bound for both sets. Any lower bound for the union must also be a lower bound for each part, which ensures that it cannot exceed the smaller infimum.

Proof. By the Existence of the Infimum Theorem, both \(\inf S\) and \(\inf T\) exist. Let \(\beta\) be the smaller of these two numbers. Since \(\beta\leq\inf S\), and \(\inf S\) is a lower bound for \(S\), every \(s\in S\) satisfies \(\beta\leq s\). Similarly, \(\beta\leq t\) for every \(t\in T\). Thus \(\beta\) is a lower bound for \(S\cup T\).

Let \(L\) be any lower bound for \(S\cup T\). Then \(L\) is a lower bound for \(S\) and for \(T\). It follows that \(L\leq\inf S\) and \(L\leq\inf T\), so \(L\leq\beta\), the smaller of those two infima. Therefore \(\beta\) is the greatest lower bound of \(S\cup T\), proving the result. \(\square\)

Infimum Versus Minimum

The infimum identifies the greatest lower bound whether or not the set contains it. A minimum, in contrast, must be an element of the set. When a minimum exists, it is automatically the infimum: it is a lower bound, and every lower bound must be less than or equal to it. Conversely, if the infimum belongs to the set, its lower-bound property makes it a minimum.

A common error is to assume that because every element lies above the infimum, some element must equal it. The set \(A=(0,6)\) shows why that conclusion is false: \(0\) is its infimum, but every element of \(A\) is strictly greater than \(0\). The Approximation Characterization captures what is guaranteed instead: elements occur less than \(\varepsilon\) above the infimum for every positive \(\varepsilon\).

The infimum is useful whenever a set has a meaningful lower edge, including when that edge is not attained. Completeness guarantees the edge for every nonempty set bounded below, and the approximation test helps verify its value without checking all lower bounds one by one.