Proving there exists a real number between any two real numbers

[naes213]

Homework Statement

If x and y are arbitrary real numbers with x<y, prove that there is at least one real z satisfying x<z<y

Homework Equations

The Attempt at a Solution

The problem arises from my inexperience in rigorously proving anything. If possible a general explanation of where to begin when trying to prove something rigorously would be more helpful than just the answer. I find that proofs generally seem obvious after i see them completed, but i sit for hours staring at them not knowing where to even remotely begin. Any help would be greatly appreciated. Thanks!

 

[morphism]

There is no general algorithm for attacking problems. But usually it helps to think about what the problem is really asking. For instance, for the problem you've posted, think about this: can you find a number between 0 and 1? In general, what's the natural choice for something that sits in between two other things? (Maybe something that's 'in the middle'?)

 

[naes213]

Ah! See, i feel really dumb now...haha...thanks. Hmmm...if i were to say something like, if x<y then (1/2)x is also less than y... would i then need to prove that? Or is that "obvious" enough to just simply state? I have a feeling that stating something as obvious is blasphemous in mathematics.

 

[morphism]

But is (1/2)x always greater than x?

 

[naes213]

Alright...thanks again! I think its just a matter of not over complicating things and thinking things through before trying to prove anything. Thanks!