Bounded above

Definition

A sequence is said to be bounded above if there exists a real number M such that every term in the sequence is less than or equal to M. The smallest such M is called the least upper bound or supremum.

5 Must Know Facts For Your Next Test

1. A sequence can be bounded above even if it does not converge.
2. If a sequence has an upper bound, then all its terms are located below this bound on the real number line.
3. The least upper bound (supremum) of a sequence is unique if it exists.
4. Boundedness above does not imply boundedness below; these are independent properties.
5. For any given sequence, verifying it is bounded above often involves finding a specific value that satisfies the condition for all terms.

Review Questions

• What does it mean for a sequence to be bounded above?
• How do you determine if a given sequence is bounded above?
• Can a sequence be both bounded above and divergent? Provide an example.

Related terms

Bounded Below: A sequence is said to be bounded below if there exists a real number m such that every term in the sequence is greater than or equal to m.

Supremum: The least upper bound of a set or sequence; the smallest value that bounds all elements from above.

Convergence: A property of sequences where terms approach a specific value as they progress towards infinity.