LUB4:  Fractions Immediately Surrounding Any Real Number

"For any [real number]  x  and positive integer  N,  there is an integer  n  such that  [Graphics:../Images/index_gr_247.gif]."  (see ita-2-3-14-pg26)

"To show this we merely have to apply LUB3 to the number  Nx,  getting an integer  n  such that  [Graphics:../Images/index_gr_248.gif]."  (see ita-2-3-14-pg26)

Note that we can't use O3 to get from Rosenlicht's intermediate result,  [Graphics:../Images/index_gr_249.gif],  to the final desired result (by multiplying both sides of the inequalities by  1/N) because we are missing the greater than zero condition.  Instead, please see my first reading of LUB4 here.

ita


Converted by Mathematica      December 4, 2007