Proving X + Y / 2 is Between X & Y in R

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In summary, the conversation is about proving the existence of a real number Z that is between two given real numbers, X and Y. One person suggests using the average of X and Y, but is unsure how to tie it in. Another person suggests adding something to both sides of the inequality, and the first person realizes that X + Y / 2 is not necessarily between X and Y.
  • #1
moo5003
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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.
 
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  • #2
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?
 
  • #3
Dunno about that (you ought to bracket things up), but if x<y why don't you just add something to both sides?
 
  • #4
matt grime said:
(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!
 

What is the formula for proving X + Y / 2 is between X and Y in R?

The formula is (X + Y) / 2.

How do you use the formula to prove X + Y / 2 is between X and Y in R?

To use the formula, you simply plug in the values for X and Y and then solve for the result. If the result is between X and Y, then the formula is proven to be true.

Can this formula be applied to any two numbers in the set of real numbers (R)?

Yes, this formula can be applied to any two numbers in the set of real numbers (R). It is a universal formula that works for all real numbers.

Is there an alternative method for proving X + Y / 2 is between X and Y in R?

Yes, there are multiple methods for proving this formula. One alternative method is using the properties of inequalities and manipulating the formula to show that it is always between X and Y.

Why is it important to prove X + Y / 2 is between X and Y in R?

It is important to prove this formula because it is a fundamental property of real numbers. It helps to establish the relationship between numbers and can be used in various mathematical calculations and proofs.

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