How to prove (X + Y) / 2 is strictly between X and Y

[moo5003]

Problem:

"Suppose X, Y in R (Real Numbers), X < Y prove there exists Z in R such that X < Z < R."

I'm currently trying to prove that X + Y / 2 satisfies this but I'm getting stuck. I first show that X + Y / 2 cannot be = to either X or Y. I then try to show that X + Y / 2 is > X since X < Y but I cannot seem to tie this in. Any help would be appreciated.

 

[moo5003]

Nvm I think I got it.

Since X < Y
Y - X must be in Positive Reals
2 is in Positive Reals thus 1/2 is in Positive Reals
Y - X / 2 is thus in Positive Reals
You can then rearrange such that you can the equation

X + Y / 2 - X in positive reals
Thus X < X + Y / 2.

Right?

 

[matt grime]

Dunno about that (you ought to bracket things up), but if x<y why don't you just add something to both sides?

 

[HallsofIvy]

(you ought to bracket things up)

In case you missed it, matt's point is: x+ y/2 is NOT between x and y:
(x+ y)/2 IS!