(1) I'm supposed to prove that$\displaystyle \sqrt{p} $ is irrational if p is prime.

I did the following

$\displaystyle p = \frac{a^2}{b^2}$

$\displaystyle b^2p=a^2$

Which means $\displaystyle a^2$ and a are factors of p but p is irrational.

Is that enough for the proof ?

(2)I'm supposed to prove the theorem

If a nonemepty set S of real numbers is bounded below, then inf S is the unique real number $\displaystyle \alpha$ such that

(a) $\displaystyle x\geq\alpha$for all x inS

(b) if $\displaystyle \epsilon>0$ , there is an $\displaystyle x_{0}$in S such that $\displaystyle x_{0}<\alpha +\epsilon$

They gave me the set T = {x|-x $\displaystyle \exists$ S}

MY answer

I said that if T is bounded from above then $\displaystyle x\leq\beta$

and $\displaystyle x_{0}>\beta+\epsilon$

so supT =$\displaystyle \beta$

If T is bounded from above then S must be bounded from below because T = {x|-x \exists S}

Let x'=-x $\displaystyle \exists S$

$\displaystyle x'\geq-\beta$

$\displaystyle \epsilon>0, x'_{0}< -\beta +\epsilon$

Suppose there is an $\displaystyle \alpha$ such that $\displaystyle -\beta<\alpha$ which is a lower bound for S

Taking $\displaystyle \epsilon=\alpha-(-\beta)$

$\displaystyle x'_{0}<-\beta+\alpha-(-\beta)$ means that

$\displaystyle x'_{0}<\alpha$ which means that $\displaystyle \alpha$ is not lower bound of S.

Therefore theorem1.1.8 is true. And there is no number greater than beta that satisfies the conditions

Is this enough ?

(3)It says to show that inf S $\displaystyle \leq$ supS

If S is a nonempty set of real number, and give sufficient conditions for equality

I said......................

If inf S exists $\displaystyle \forall x \exists S$

$\displaystyle x \geq \alpha=infS$

AND

if sup exists then $\displaystyle \forall x \exists S$

$\displaystyle x\leq\alpha=supS$

Case1

$\displaystyle x>\alpha$ and $\displaystyle x<\beta \forall x$

then according to the transitive property

$\displaystyle \alpha<beta$ which implies $\displaystyle infS<supS$

Case2

$\displaystyle x=\alpha$ and $\displaystyle x<\beta \forall x$

then it follows again for the transitivity of real numbers that

$\displaystyle \alpha<\beta$

Case3

$\displaystyle x= \alpha$ and $\displaystyle x= \beta \forall x$

then it follows that

$\displaystyle \alpha = \beta$

So the inequality

$\displaystyle infS \leq supS$ is satisfied.

MY major problem is i'm not sure if this suffices as proofs or not.