What are the two criteria for a number to be the supremum of a set?

Conditions

• The set S is non-empty.
• β is a real number.

Reasoning, step by step

1. Verify that β is an upper bound for S (i.e., for all x in S, $x \le β$).
2. Verify that β is the least upper bound (i.e., for any other upper bound M, $β \le M$).
3. Conclude that β is the supremum if both criteria are met.

Example

The video presents the formal mathematical property for β being an upper bound: for every element x in S, x must be less than or equal to β. It then introduces the second criterion by showing that any value smaller than β fails to be an upper bound.

Common misconceptions

• Thinking that the supremum must be an element of the set S.
• Confusing the supremum with the maximum value of a set.

Watch the explanation

Explore next

• How do you prove that a number is the least upper bound?
• What is the logical structure of proving minimality via contradiction?