Calculator
Formula used
For a set S, a number m is the least upper bound when two conditions hold. First, every element of S is less than or equal to m. Second, for every positive epsilon, some element of S lies within epsilon of m from below.
Definition: m = sup(S) if x ≤ m for all x in S, and for every ε > 0 there exists x in S with m − ε < x ≤ m.
Finite sets: sup(S) = max(S). Intervals: sup(a, b) = b, sup[a, b) = b, sup[a, b] = b, and sup(a, b] = b whenever b is finite.
If the set is empty or unbounded above, this calculator reports that a least upper bound is not defined in the real numbers.
How to use this calculator
- Choose Finite set for listed numbers or Interval for endpoint notation.
- Enter numbers with commas, spaces, or semicolons, or supply the interval endpoints.
- Select open or closed endpoints when using interval mode.
- Add an optional candidate upper bound to test whether it is least.
- Set precision and epsilon, then click Calculate Least Upper Bound.
- Use the CSV and PDF buttons to export the current result summary.
Example data table
| Case | Input | Type | Least upper bound | Maximum attained | Note |
|---|---|---|---|---|---|
| 1 | {2, 4, 9, 9.5} | Finite set | 9.5 | Yes | The largest listed value is the supremum. |
| 2 | (0, 10) | Interval | 10 | No | The right endpoint is excluded but still gives the supremum. |
| 3 | [3, 3] | Interval | 3 | Yes | A singleton interval has one element, so max equals sup. |
| 4 | [1.2, 5.7) | Interval | 5.7 | No | Open right endpoints are not maxima. |
| 5 | (-2, Infinity) | Interval | Not defined | No | The set is unbounded above. |
FAQs
1. What is a least upper bound?
It is the smallest number that is still greater than or equal to every element in a set. In real analysis, it is also called the supremum.
2. Is the least upper bound always inside the set?
No. For open intervals like (0, 1), the least upper bound is 1, but 1 is not part of the set.
3. When does the least upper bound equal the maximum?
They are equal when the supremum belongs to the set. Every non-empty finite set has this property because its largest element is included.
4. Can an unbounded set have a least upper bound?
Not in the real numbers. If values can grow without limit, no real number can serve as the smallest upper bound.
5. Why does an open interval still have a supremum?
Because the right endpoint can still bound all elements from above, even when the endpoint itself is excluded from membership.
6. What does the epsilon witness mean?
It shows a set element that lies very close to the supremum from below. This helps illustrate the defining property of least upper bounds.
7. Why test a candidate upper bound?
It helps compare your chosen bound against the true supremum. The calculator shows whether the candidate works and whether it is least.
8. Can I export the result?
Yes. The page can generate a CSV summary for spreadsheets and a PDF summary for quick sharing or printing.