Let b∈R, b>0 and let B:={x^2 | x ∈(−b, 0) }.Find Sup(B),if it exists. Make sure to fully prove your assertion.

Question

Let b∈R, b>0 and let B:={x^2 | x ∈(−b, 0) }.Find Sup(B),if it exists. Make sure to fully prove your assertion.

Answer 1

B = (0, b^2) for x = (-b,0) for function y=x^2 which is continuous.

For real a, 0 < a < b, then -b < -a < 0 .

So 0 < a*a < a*b < b*b

0 < a^2 < ab < b^2

So then b^2 is an upper bound.

Proceeding by contradiction

Suppose there is ANOTHER upper bound of set B, namely C, distinct from b, that is less than b.

Then there exists c in (-b,0) such that c^2 is greater than any element of the range of f(x)

That is -c < -b < 0 which forces c^2 > b^2 > 0 which implies that b^2 is not an upper bound.

Contradiction

Therefore b^2 = c^2 since the LUB must be unqiue

Answer 2

When -b < x < 0, x² takes on all values between 0 and b² (not including 0 and b²).

B = {x² l x ∈ (-b, 0) where b > 0}

sup(B) = least upper bound of B = b²