What is the definition of an upper bound for a non-empty set of real numbers?

Conditions

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

Reasoning, step by step

1. Identify the set S and the candidate upper bound M.
2. Check the condition for every element x in S.
3. Verify that $x \le M$ holds for all elements.
4. Conclude that M is an upper bound if the condition is satisfied.

Example

In the video, three horizontal dashed lines labeled M1, M2, and β are drawn above a scatter plot of set S. All data points lie below these lines, confirming that M1, M2, and β are all valid upper bounds for the set S.

Common misconceptions

• Believing that an upper bound must be an element of the set S.
• Thinking that only one upper bound can exist for a given set.

Watch the explanation

Explore next

• What is the supremum of a set?
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