Supremum and Infimum of Bounded Sets Multiplication

[esuahcdss12]

Hey all,
I started to learn this subject, and i understtod how to find the supremum and infimum of a given set or function.
but I have problem with one question which I can not solve, and I don't know how to start.
This is the quesion:

Given to bounded sets X and Y, which their element are REAL and non negative numbers,
so that X*Y = {x*y: x in X, y in Y}
prove that:

a)inf{X*Y} = infX *infY
b)sup{X*Y}= supX * supY

Can anyone help please?

 

[Euge]

Welcome, esuahcdss12! (Wave)

Let's start with question a). Let $a = \inf X$ and $b = \inf Y$. Given $x\in X$ and $y\in Y$, $x \ge a$ and $y\ge b$, so that $xy\ge ab$ (which follows from the fact that $x$ and $y$ are nonnegative). Deduce that $\inf(XY) \ge ab$. To obtain $\inf(XY) \le ab$, note that if $\epsilon > 0$, $a + \epsilon$ is not a lower bound for $X$ and $b + \epsilon$ is not a lower bound for $Y$. So there are $x\in X$ and $y\in Y$ such that $a + \epsilon > x$ and $b + \epsilon > y$. Hence $(a + \epsilon)(b + \epsilon) > xy \ge \inf(XY)$. Since $\epsilon$ was arbitrary chosen, deduce that $ab \ge \inf(XY)$.

The argument for question b) is similar.