154693

1 We need to prove that x+y<1+xy for all values of x and y provided that
2 0<x<1 and 0<y<1
3 let x=a/b and y=c/d , where a<b and c<d due to the conditions in line 2
4 (a/b)+(c/d)<1+(ac/bd)
5 (ad+bc)/(bd)<1+(ac)/(bd)
6 (ad+bc-ac)/(bd)<1
7 ad+bc-ac<bd
8 a(d-c)<b(d-c)
9 a<b
10 Line 9 is true because of line 2, therefore
11 x+y<1+xy for all x and y between and not including 0 and 1