Question

how to prove that:
let x,y be two real no., for some rational no.z
x 13 years ago

Answer 1

you should use two theorems,"Archimedes properties"and"minimum natural number theorem "

let's show them.

suppose x<y,
by"Archimedes properties",we know "exist a integer n ,make n(y-x)>1"

and them by "minimum natural number theorem" ,we know any elements of set of integer,either greater than nx or smaller than nx,when nx is not a integer.so,we get a minimum number "m>nx",make "(m-1)<nx<m".
and,if nx is a integer,then (m-1)<=nx<m.

so nx<=m<nx+1<ny. nx<=m<ny,x<=m/n<y. m,n are belong to integer.

by the way ,my english is very poor.