Is There Always a Real Number Between Two Arbitrary Real Numbers?

[courtrigrad]

Hello all

I need help with the following proofs

1. If x and y are arbitrary real numbers, prove that there is at least one real z satisfying x < z < y. (Do I just use the Archimedian Property?)

Thanks

 

[hypermorphism]

(Do I just use the Archimedian Property?)

Yep.

 

[matt grime]

No. and it isn't true as stated.

x and y must be distinct, and the requirement of being R is unnecessary, since it is true for Q (and C).


what is (x+y)/2?

 

[hypermorphism]

Good catch. I was thinking of rational numbers between arbitrary reals.