Answer to Question #4227 in Real Analysis for Shiro

Question

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.

Answer 1

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