# Simple supremum question

• Sep 28th 2009, 03:28 PM
cgiulz
Simple supremum question
If $A$ and $B$ are bounded and non-empty subsets of $\mathbb{R}$, and $c \in \mathbb{R}$, let

$cA =$ { $ca: a \in A$}.

Prove the following:
if $c > 0$, supremum $cA = c$ supremum $A$.

[SOLVED]
• Sep 28th 2009, 06:59 PM
Danneedshelp
Quote:

Originally Posted by cgiulz
If $A$ and $B$ are bounded and non-empty subsets of $\mathbb{R}$. Prove the following:

if $c > 0$, supremum $cA = c$ supremum $A$.

Let $\alpha=sup(A)$. Since $\alpha$ is an upper bound for $A$, $c\alpha\geq\\ca$ for all $a\in{A}$. Therefore, $c\alpha$ is an upper bound for $A$. Moreover, if we let $\beta$ be any other upper bound for $cA$, we have that $\frac{\beta}{c}$ is an upper bound for $A$. Thus, $\alpha\leq\frac{\beta}{c}$ $\Leftrightarrow\\c\alpha\leq\beta$. So, $c\alpha$ satisfies the definition of $sup(cA)$.

That is my stab at it.
• Sep 29th 2009, 12:57 PM
cgiulz
Wow, I left out a piece sorry! (Itwasntme)