# Proving that a set is not bounded from above.

• Skirdge
In summary, to prove that a set {a, a^2, a^3, ...} is not bounded from above when a is a real number greater than 1, we first need to find a positive integer n such that a > 1 + 1/n. This can be achieved by showing that a-1 is a positive real number and thus there exists an integer 1/n such that a-1 > 1/n. Then, to prove that a^n > (1 + 1/n)^n >/= 2, we can look at the sequence (a^(nk)) as k varies and show that it is not bounded above. This can be done by comparing it to the sequence {2,
Skirdge

## Homework Statement

Prove that if a is a real number, a > 1, then the set {a, a^2, a^3, ...} is not bounded from above. Hint: First find a positive integer n such that a > 1 + 1/n and prove that a^n > (1 + 1/n)^n >/= 2.

## The Attempt at a Solution

Showing that there exists a positive integer n such that a > 1 + 1/n is not difficult. Since a > 1, a-1 is a positive real number so there exists an integer 1/n such that a-1 > 1/n and thus a > 1 + 1/n. Proving the second set of inequalities is not difficult either. I'm at a complete loss as to how the hint relates to the problem.

Look at the sequence ##(a^{nk})## as ##k## varies.

Can you prove that {2. 22, 23, 24, ...} is not bounded above?

## 1. How do you define a set that is not bounded from above?

A set is not bounded from above if there is no maximum element or if the maximum element is infinite. This means that the set can continue to increase without limit.

## 2. What is the difference between a bounded and unbounded set?

A bounded set has both a maximum and minimum element, while an unbounded set has either no maximum element or an infinite maximum element.

## 3. How can you prove that a set is not bounded from above?

To prove that a set is not bounded from above, you can show that for any given upper bound, there exists an element in the set that is greater than that upper bound. This would demonstrate that the set can continue to increase without limit.

## 4. Can a set be both bounded and unbounded?

No, a set can only be either bounded or unbounded. It cannot be both at the same time.

## 5. Are there any real-life examples of an unbounded set?

Yes, an example of an unbounded set in real life is the set of all positive integers. This set has no maximum element and can continue to increase without limit.

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