Preliminary:  Upper and Lower Bounds

Let  S  be a nonempty set of real numbers.  As I look at the inequalities below, I find it helpful to imagine a ruler which is in front of me with the smaller numbers on the left and the larger numbers on the right.

A real number  a  is a lower bound of  S  if:    for all  s  in  S,    [Graphics:../Images/index_gr_238.gif].

A real number  b  is an upper bound of  S  if :    for all  s  in  S,    [Graphics:../Images/index_gr_239.gif].

The set  S  would be represented by a strip of paper on the ruler.  a  is to the left of the paper or at the left edge of the paper.  b  is to the right of the paper or at the right edge of the paper.  The numbers  s  are represented by points on the paper.  In the following negations of the above statements, note that   for all   becomes   there exists.   For clarity, suppose that  a  and  b  are still lower and upper bounds respectively.

If a real number  x  is not a lower bound for  S,  then    there exists an element  s in  S  such that  [Graphics:../Images/index_gr_240.gif].

If a real number  y  is not an upper bound for  S,  then    there exists an element  s  in  S  such that  [Graphics:../Images/index_gr_241.gif].

On the ruler described above:  x  can be any point to the right of the left edge of the paper, but it can't be at the left edge.  y  can be any point to the left of right edge of the paper, but it can't be at the right edge.

Let  α  (alpha) be the greatest lower bound, and let  β  (beta) be the least upper bound.  α  is at the left edge of the paper, and  β  is at the right edge of the paper.


Converted by Mathematica      December 4, 2007