How to Find Suprema and Infima Without Proof

A supremum or infimum is a boundary value of a set: the supremum is its least upper bound, and the infimum is its greatest lower bound. The boundary value does not have to be an element of the set. That distinction is why a set can have a supremum but no maximum, or an infimum but no minimum.

The title “Abbott 1.3.8” does not include the exercise’s set definitions, and exercise numbering can vary between editions. So there is no unique list of numerical answers to give from the title alone. The method below explains how to find the answers once you have the sets; the examples show the main cases.

The definitions to use

For a nonempty set $A\subseteq\mathbb{R}$, a real number $u$ is an upper bound for $A$ if every $a\in A$ satisfies $a\leq u$. If there is an upper bound, the supremum $\sup A$ is the smallest one. Similarly, $l$ is a lower bound if $l\leq a$ for every $a\in A$, and the infimum $\inf A$ is the largest lower bound.

If the set contains its supremum, that supremum is also its maximum. If it contains its infimum, that infimum is also its minimum. Otherwise, the set has no maximum or minimum at that boundary. A set may be unbounded above, unbounded below, or both; in those cases it has no corresponding finite real supremum or infimum.

A quick method for an exercise

For each set, work through these checks:

1. Read the membership condition carefully. Note whether endpoints are included and what values the variables may take.
2. Find candidate edges. For intervals, these are usually endpoints. For sequences or expressions, look at the largest and smallest values, including values approached as an index grows.
3. Check that the candidates really bound the set. Every element must be at most the proposed supremum and at least the proposed infimum.
4. Check whether the bounds are attained. Ask whether the candidate itself satisfies the set’s membership condition.

“Without proof” usually means you can state the values without writing a formal verification. It does not change the definitions: you still need to distinguish a bound from the least upper bound, and to check endpoint inclusion.

Worked example: a sequence of reciprocals

Let

$$A=\left\{\frac{1}{n}:n\in\mathbb{N}\right\},$$

using the convention $\mathbb{N}=\{1,2,3,\ldots\}$. Its elements begin $1,\frac12,\frac13,\frac14,\ldots$. Each is positive and at most $1$, and $1$ is in the set (take $n=1$). Therefore

$$\sup A=1,\qquad \max A=1.$$

The elements get arbitrarily close to $0$, but none equals $0$. Thus $0$ is the greatest lower bound, but not a minimum:

$$\inf A=0,\qquad A\text{ has no minimum}.$$

This example shows why the infimum is not necessarily the smallest element. If a text instead defines $\mathbb{N}$ to start at $0$, this particular expression is undefined at $n=0$, so check the convention or the exercise’s stated domain.

Worked example: an open interval described by a condition

Consider

$$B=\{x\in\mathbb{R}:x^2<2\}.$$

The inequality is equivalent to $-\sqrt{2}<x<\sqrt{2}$, so $B=(-\sqrt{2},\sqrt{2})$. Its boundary values are therefore

$$\inf B=-\sqrt{2},\qquad \sup B=\sqrt{2}.$$

Neither boundary value belongs to $B$, because squaring either one gives $2$, not a number strictly less than $2$. Hence $B$ has neither a minimum nor a maximum. The same supremum and infimum result if the set is restricted to rational $x$: rational numbers occur arbitrarily close to each endpoint from inside the interval, but neither endpoint is rational.

Common mistakes

• Calling a bound the supremum without checking it is least. If $5$ is an upper bound, a smaller upper bound might exist. The supremum is the smallest upper bound, not just any upper bound.
• Assuming the endpoint is included. The sets $[0,1]$ and $(0,1)$ have the same infimum and supremum, but only the first has a minimum and maximum.
• Confusing “approaches” with “equals.” A sequence of set elements can get as close as desired to a boundary without ever reaching it. That boundary may still be the infimum or supremum.
• Ignoring unboundedness. For example, $\{x\in\mathbb{R}:x>0\}$ has infimum $0$ but no finite real supremum.
• Forgetting conventions or domains. Whether $0$ is in $\mathbb{N}$, and whether a set is taken in $\mathbb{Q}$ or $\mathbb{R}$, can matter.

When writing a short answer, a useful format is: “$\inf A=\ldots$ and $\sup A=\ldots$; the infimum [is/is not] attained, and the supremum [is/is not] attained.” To give exact answers to a particular part of Exercise 1.3.8, use the set definitions printed in your edition.