1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Proving that a set is not bounded from above.

  1. Jan 12, 2013 #1
    1. The problem statement, all variables and given/known data
    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.


    2. Relevant equations



    3. 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.
     
  2. jcsd
  3. Jan 12, 2013 #2
    Look at the sequence ##(a^{nk})## as ##k## varies.
     
  4. Jan 12, 2013 #3
    Can you prove that {2. 22, 23, 24, ...} is not bounded above?
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook