Proving that a set is not bounded from above.

[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.

Homework Equations

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.

 

[gopher_p]

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

 

[Joffan]

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