Question

X, Y, Z, JER. XKY Prove Z <# X AND Y<j, if and only if (x,y) [z,j]

Prove

Answer 1

X Y Z JER Such that X<Y. Let Z <x and tsj. X<Y then theres exust any rEr such that Z <x xoy Z, JER. and acr. let and raj. and x <Y. now consider ire (x,y) the open mterval in (x,y) element lie between X and Y. ie for all let ४ be i.e any element from (x,x). rt (x,x) .x<r<8 z<x<r<y sj De (2,5). re [z.j] this implies (X,) E [2] (x,y) s [z,j]. [ab]={x, asa<b} since is arbitry in TA り so conversely, let X<Y and (X,Y) [Z ). ret 7 E(X,Y) メ<r<Y. again (x,y) [2, 3] re [z,j] » then Z srs J eo element be arbitrary R. let then x < XtE

and (X +) E (x,y) E [2,3] [z, j] x+EE ☺ -) Z < X+€ s -) Z <x+€. this above gives ZsX since e is arbitray so ZEX similarly Y-Ear and Y-E € (x,y) s Ezf] > (Y-E) E [Z,j] -) Z <CY-€) <j > Y-es => Y Zjte. &is arbitrary therefore ysj. if X, Y, Z,) ER so and er then zex and ups) y and only if yxz