Let B={-x : x ϵ A}, A a non-empty subset of R. Show B is bounded below

Hi, I have the following problem... I know the second bit of it, but I have no clue how to prove the first bit. I know B is bounded below (it is basically set A but with opposite signs, so the supremum of A will be the lowest number of set B, its infimum), but I don't know how to express it mathematically. Thanks for any help you can give me!

"Let A be a non-empty subset of R that is bounded above. Let

B = {-x : x ϵ A}.

Show that B is bounded below. How are sup A (supremum of A) and inf B (infimum) related?"

I don't know how to prove that B is bounded below, but for the question "How are sup A (supremum of A) and inf B (infimum) related?", I have the following:

sup A = -inf B.

Re: Let B={-x : x ϵ A}, A a non-empty subset of R. Show B is bounded below

Quote:

Originally Posted by

**juanma101285** "Let A be a non-empty subset of R that is bounded above. Let

B = {-x : x ϵ A}.

Show that B is bounded below. How are sup A (supremum of A) and inf B (infimum) related?" sup A = -inf B.

If $\displaystyle \alpha=\sup(A)$ then if $\displaystyle t\in B$ then $\displaystyle -t\in A$.

So $\displaystyle -t\le\alpha$ or $\displaystyle -\alpha\le t$.

Thus $\displaystyle -\alpha\le\beta=\inf(B)$

Suppose that $\displaystyle -\alpha<\beta$ then $\displaystyle \alpha>-\beta$ so $\displaystyle \left( {\exists x \in A} \right)\left[ { - \beta < x \leqslant \alpha } \right]$

What is wrong with that?

Re: Let B={-x : x ϵ A}, A a non-empty subset of R. Show B is bounded below

Thanks for the message!

Ermm... so that would be a contradiction because that would mean x is sup(A) and -x is inf(B)??

Re: Let B={-x : x ϵ A}, A a non-empty subset of R. Show B is bounded below

Quote:

Originally Posted by

**juanma101285** Thanks for the message!

Ermm... so that would be a contradiction because that would mean x is sup(A) and -x is inf(B)??

Well it means $\displaystyle -x\in B$ and $\displaystyle -x<\beta~.$

Can that happen?

Re: Let B={-x : x ϵ A}, A a non-empty subset of R. Show B is bounded below

Let a be an upper bound of A. Let x be any member of B. Then x= -y for some y in A. Then y< a. If follows that -y= x> -a. Thus, -a is a lower bound for B.