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.
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A sequence can be bounded above even if it does not converge.
If a sequence has an upper bound, then all its terms are located below this bound on the real number line.
The least upper bound (supremum) of a sequence is unique if it exists.
Boundedness above does not imply boundedness below; these are independent properties.
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.