Why does the Least Upper Bound Property guarantee that L - ε is not an upper bound for the set of sequence values?

Conditions

• $L$ is the supremum (least upper bound) of the set $\{a_n\}$.
• $\epsilon > 0$ is an arbitrary positive real number.

Reasoning, step by step

1. Define $L$ as the supremum of the set of sequence values.
2. Note that $L - \epsilon < L$ because $\epsilon > 0$.
3. Apply the definition of supremum: no number smaller than $L$ can be an upper bound.
4. Conclude that $L - \epsilon$ fails to be an upper bound.
5. Infer that there exists some term $a_N$ such that $a_N > L - \epsilon$.

Example

The script explains: 'Because L is the *least* upper bound, L - ε cannot be an upper bound, implying some term $a_N$ exceeds L - ε.'

Common misconceptions

• Thinking that $L - \epsilon$ might still be an upper bound if the sequence doesn't reach close to $L$.
• Confusing the supremum with a maximum; the supremum need not be attained by any element, but elements must get arbitrarily close to it.

Watch the explanation

Explore next

• Does the sequence necessarily contain the value L?
• What if the sequence is bounded above by multiple numbers? Which one is L?