Prove Rational Numbers Have Midpoint: x,y ∈ Q

[kaos]

Can someone check if my proof is correct.Please exscuse the bad notation, I've no idea how to type the symbols.
The question was prove that between any 2 rational number , there is a third rational.

x,y ,z are elements of Q
(for all x ) (for all y) (there exist z)[x>z>y] <->
(for all x ) (for all y) (there exist z)[(x>z) ^ (z>y)]

Proof by contradiction:
Suppose its false that for any x and y , there exists a z between x and y

~((for all x ) (for all y) (there exist z)[x>z>y])
(there exists x) (there exists y)( for all z)[ (x< or = z) V (z < or = y)]
There is no x that is smaller than or equals to any z.
There is no y that is larger than or equals to any z.
Both are false, the disjunction is false.
Therefore the statement (there exists x) (there exists y)( for all z)[ (x< or = z) V (z < or = y)]
is false and the statement (for all x ) (for all y) (there exist z)[x>z>y] is true.

 

[haruspex]

Since you have not used any facts about rational numbers, it seems vanishingly unlikely that your proof is valid.
How about doing something really simple and obvious: given two rationals p/q and r/s construct a rational that lies between them.

 

[kaos]

If i construct a rational in between p/q and r/s , i doesn't apply to any other rationals, so it doesn't really prove anything. Am i misinterpreting your statement ( I am really bad at math so please excuse my lack of ability)?

 

[haruspex]

If i construct a rational in between p/q and r/s , i doesn't apply to any other rationals, so it doesn't really prove anything. Am i misinterpreting your statement ( I am really bad at math so please excuse my lack of ability)?

P, q, r and s can be any integers (q, s nonzero). If you construct a rational between p/q and r/s then you will have provided a general construction for any given pair of rationals.

 

[Ibix]

p/q and r/s are arbitrary rational numbers. Haruspex is suggesting that you construct an expression in terms of p, q, r and s that is rational and guaranteed to lie between the the two.

 

[kaos]

Ah ok i see , thanks guys.