Welcome to our community

Be a part of something great, join today!

A subset E of R is bounded above if and only if the set -E is bounded below

G-X

New member
Jul 11, 2020
21
a) Suppose the set $-E = ${$-x : x \in E$} is bounded below. For $-E$ to be bounded below that means there $\exists w \in R$ s.t. $\forall y \in E, w \le y$.

We want to show that $E \subset R$ is bounded above meaning that $\forall x \in E, x \le u$. By definition of $-E$ and bounded below, we can imply that $y = -x$.

We can change the equality such that $x = -y$ and $-y \le -w$. By substitution, $x \le -w$. If we let $-w = u$ then $x \le u$ and thus $E \subset R$ is bounded above.

b) Suppose that $E \subset R$ is bounded above. For $E$ to be bounded above that means there $\exists u \in R$ s.t. $\forall x \in E, x \le u$.

We want to show that the set $-E = ${$-x : x \in E$} is bounded below meaning that $\forall y \in -E, w \le y$. By definition of $-E$ and bounded below,

we can imply that $y = -x$. By changing the equality of bounded above, we can show that $-u \le -x$. By substitution of $y = -x$, we rewrite that

$-u \le y$. If we let $w = -u$ then $w \le y$ and thus we have shown that $-E$ is bounded below.