# Real analysis

• Jun 28th 2013, 12:02 AM
Mondella
Real analysis
P/s i need help to prove that every positive real no has a unigue positive square root
• Jun 28th 2013, 06:01 AM
Plato
Re: Real analysis
Quote:

Originally Posted by Mondella
P/s i need help to prove that every positive real no has a unigue positive square root

Any proof of this depends upon the set of definitions and axioms with the sequence of theorems you have.

Here is a general approach. If $\displaystyle c\in\mathbb{R}^+$ prove that if $\displaystyle 0<t~\&~t^2=c$ then there is at most one such $\displaystyle t$.
Let $\displaystyle E=\{x\in\mathbb{R}^+:x^2<c\}$. You want to show that $\displaystyle E\ne\emptyset$ and $\displaystyle E$ is bounded above.
If you can show those then use the completeness property.

Hint: let $\displaystyle t=\frac{c}{1+c}$. Is it true that $\displaystyle t\in E~?$

What can you do with that?
• Jun 28th 2013, 11:50 AM
Lord Voldemort
Re: Real analysis
Plato, can you explain why this doesn't work? Given distinct real numbers x and y in R, suppose they have the same square root z. By definition, z = sqrt(x) and z=sqrt(y), so we have z^2=x=y which is a contradiction, so two distinct real numbers can not have the same square root.
• Jun 28th 2013, 01:11 PM
Plato
Re: Real analysis
Quote:

Originally Posted by Lord Voldemort
Plato, can you explain why this doesn't work? Given distinct real numbers x and y in R, suppose they have the same square root z. By definition, z = sqrt(x) and z=sqrt(y), so we have z^2=x=y which is a contradiction, so two distinct real numbers can not have the same square root.

How do does that prove that if $\displaystyle x\in\mathbb{R}^+$ that $\displaystyle \sqrt{x}$ exists at all?