Answer to Question #267513 in Real Analysis for jeff

Question

x is a member of A

Answer 1

Incomplete question:

Let us take an example related to given statement:

Suppose A is bounded and B={x² | x is a member of A}. Show if sup(A)=$\alpha$  then sup(B) = $\alpha$

Solution:

Given that A is bounded sub set of $\mathbb{R}$ and $\sup A=\alpha$ and $B=\left\{x^{2} / x \in A\right\}$

Since $x \leq \alpha \forall x \in A$

 $\begin{aligned} &x^{2} \leq \alpha^{2} \\ &x^{2} \leq \alpha^{2} \forall x^{2} \in B \end{aligned}$

Therefore $\alpha^{2}$ is upper bound for B

Let $\beta$ be an other upper bound for B

so that $x^{2} \leq \beta \forall x^{2} \in B$

$\Rightarrow x \leq \sqrt{\beta} \forall x \in A$

Therefore $\sqrt{\beta}$ is upper bound for A.

Since $\alpha$ is suprimum of A, we have $\alpha \leq \sqrt{\beta} \Rightarrow \alpha^{2} \leq \beta$ .

Therefore $\alpha^{2}$ is least upper bound for B

Hence sup B $=\alpha^{2}$