LUB3:  Integers Immediately Surrounding Any Real Number

"For any [real number]  x  there is an integer  n  such that  [Graphics:../Images/index_gr_244.gif]."  (see ita-2-3-13-pg26)

"To prove this, choose an integer  N>|x|,  so that  -N<x<N.  The integers from  -N  to  N  form the finite set  {-N,...,0,1,...N}  and all we need do is take  n  to be the greatest of these that is less than or equal to x."  (see ita-2-3-13-pg26)

As a preliminary step towards proving Rosenlicht's first sentence immediately above, please see the page here on the negation of an inequality, by which I mean multiplying both sides of an inequality by  -1.  Note the similarity of the following proof with the  only if  proof in the subsection on  ε-neighborhoods  under  Absolute Values  in the  Order Property  section.  Also recall that the  if and only if  logical statement was presented in the  Preliminary  part of the  Order Property  section.


For the  'take n to be the greatest of these'  part of Rosenlicht's proof, please scroll down my first reading of LUB3 here.


Converted by Mathematica      December 4, 2007