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.

[Graphics:../Images/index_gr_245.gif]
[Graphics:../Images/index_gr_246.gif]

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

ita


Converted by Mathematica      December 4, 2007