Theorem 1.21 Rudin. Obviously wrong stated, right?

[saim_]

Theorem 1.21 Rudin. Obviously wrongly stated, right?

Theorem 1.21 in Rudin states:

For every real $x > 0$, and every integer $n > 0$, there is one and only one real $y$ such that $y^{n} = x$.

The bold part should be "only one positive real", shouldn't it, or am I missing something? The proof also start with with an implicit assumption that $y$ is positive.

 

[superg33k]

Yeah I agree it should be positive.

The obvious counter example if it wasn't is n=2 then both -1^2=1 and 1^2=1 making the statement false.

 

[GenePeer]

I'm looking at the Third Edition and it says "one and only one positive real."

 

[saim_]

Thanks guys.

@GenePeer: I got third edition as well and it says exactly what I wrote. The error must be only in some prints then.

 

[CellCoree]

I would respond to this by saying that while the statement may seem obvious and straightforward, it is important to carefully consider all aspects and assumptions made in a theorem before concluding that it is obviously wrong. In this case, the statement may seem obvious because it is a well-known fact that every positive real number has a unique positive nth root. However, the proof of this theorem does require the assumption that y is positive, which may not be immediately apparent to everyone. It is important to be thorough and precise in mathematical proofs in order to ensure accuracy and avoid misunderstandings.