Show that if A is bounded above by the number a and set B is bounded above by th

Question

Show that if A is bounded above by the number a and set B is bounded above by the number b, then A n B (a intersect b) is bounded above by a.

Explanation / Answer

well, suppose that the upper bound of the set is u, and let us take a number n such that n > u now, the r'th term of the set is a^r let it be equal to x i.e. a^r = x taking log on both sides, r log a = log x or, r = log x / log a now if x = n, r = log n / log a (obviously n > 1, as any number greater than the upper bound of a set whose smallest element is greater than 1 has to be greater than 1) evaluating this, we see that r is a positive finite number if a > 1, infinite if a = 1, and negative if a < 1 thus, for a>1, there is a positive number r such that a^r = n but a^r (or, if r is not a whole number, a^(r+m) such that m>0 and r + m is a whole number) clearly belongs to the set {a, a^2, a^3, ...} and is the r'th term (or (r+m)th term) in it. this contradicts our previous assumption that u is the upper bound of the set, as a number greater than the upper bound cannot be in the set

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