How do you prove that a number is the least upper bound using contradiction?

Conditions

• β is an upper bound for the set S.
• γ is a real number such that $γ < β$.

Reasoning, step by step

1. Introduce a new value γ that is strictly less than β.
2. Assume for the sake of contradiction that γ is an upper bound for S.
3. Show that there exists an element x in S such that $x > γ$.
4. Conclude that γ cannot be an upper bound.
5. Deduce that β is the least upper bound (supremum) of S.

Example

The animation shows a green line γ introduced just below β. An orange point breaks through γ, proving there exists an element $x > γ$. Since any number smaller than β fails to contain the whole set, β is conclusively identified as the supremum.

Common misconceptions

• Believing that finding one element greater than γ is not enough to disprove γ as an upper bound.
• Thinking that the supremum must be the largest element in the set.

Watch the explanation

Explore next

• What is the formal definition of the supremum?
• Why is the supremum unique?