Show that a set of real numbers E is bounded if and only if there is a positive number r so that absolute value x<r for all x in e.
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Expert's answer
2011-09-18T17:45:58-0400
A set S of real numbers is called bounded from above if there is a real number k such that k ≥ s for all s in S. The number k is called an upper bound of S.A set of real numbers is bounded from below if there is a real number m such that s>=m fo all s in S. m is lower bound . A set S is bounded if it has both upper and lower bounds. 1. => set S is bounded=> it has lower and upper bounds, respectively m and k. m=>k if m,k>0 then we take r=m and |x|<r for all x in S (cause in this case x>0 and |x|=x) if m>0, k<0 then we take r=max{m, |k|} if m,k<0 r=|k| 2.<= assume |x|<r for all x in S Hence -r<x<r for all x so x>-r bounded from below x<r bounded from above
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