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Homework Statement
Let a > 1 and let S = {a, a2, a3, ...}. Prove that S is not bounded from above.
Relevant theorems
The least upper bound property of the reals and these consequences:
(1) For any real x there is an integer n satisfying n > x .
(2) For any positive real x there is a positive integer n satisfying 1/n < x.
(3) For any real x there is an integer n satisfying n ≤ x < n + 1.
(4) For any real x and positive integer N, there is an integer n satisfying n/N ≤ x < (n+1)/N.
(5) For any real x and e, there is a rational r satisfying |x - r| < e.
The attempt at a solution
I was thinking of a proof by contradiction by I can't think of anything contradictory. Any tips?
Let a > 1 and let S = {a, a2, a3, ...}. Prove that S is not bounded from above.
Relevant theorems
The least upper bound property of the reals and these consequences:
(1) For any real x there is an integer n satisfying n > x .
(2) For any positive real x there is a positive integer n satisfying 1/n < x.
(3) For any real x there is an integer n satisfying n ≤ x < n + 1.
(4) For any real x and positive integer N, there is an integer n satisfying n/N ≤ x < (n+1)/N.
(5) For any real x and e, there is a rational r satisfying |x - r| < e.
The attempt at a solution
I was thinking of a proof by contradiction by I can't think of anything contradictory. Any tips?