True/False: Let b ∈ R, b>0 and let B:={x^2 | x ∈ (−b,0)}. We then have that inf(B) = 0 and sup(B) = b^2

Question

Statement: "Let b ∈ R, b>0 and let B:={x^2 | x ∈ (−b,0)}. We then have that inf(B) = 0 and sup(B) = b^2"

I'm not sure where to start with this question. Can anyone help?

Answer 1

First let's make sure you can read what those symbols mean.

b is an element of the real numbers and b is greater than 0.

The set B is the set of all x² such that x is an element of the open interval from -b (opposite of b) to zero.

The inf of set B is also called the Greatest Lower Bound--the largest number that is less than or equal to every number in the set.

Let's make up a specific example. Let b=5 which is greater than 0. Then x belongs to the open interval (-5, 0). The set of elements in B is all the numbers in that interval squared. The smallest number in set B has to be greater than 0 because any number squared (except 0) is positive. So 0 is the greatest lower bound for set B.

Similarly the sup(B) is also called the Least Upper Bound. For the example above, where b = 5, b² is 25. And 25 has to be greater than any number in (-5, 0) squared. So b² is a least upper bound for set B.

Sounds like a true statement.