Set of upper bounds

[Baron]

Let C be a non-empty set of R. Prove that the set of upper bounds of C cannot equal (1, infinity)

My work:

I have no idea where to start. I think I have to prove by contradiction. So let C be a non-empty subset of R and assume the set of upper bounds of C is equal to (1, infinity). Now, I'm having difficulties wondering why the statement is true as if C = (0,1), the set of upper bounds of C is equal to (1, infinity) as all numbers in (1, infinity) are greater than (0,1).

So how do I start?

 

[pka]

Let C be a non-empty set of R. Prove that the set of upper bounds of C cannot equal (1, infinity)

Suppose that \(\displaystyle (1,\infty)\) is the set of upper bounds for \(\displaystyle C\).

That means that \(\displaystyle 1\) is not an upper bound for \(\displaystyle C\), why?

That means \(\displaystyle \exists x\in C\) such that \(\displaystyle x>1\). WHY?

What is wrong with that?

 

[Baron]

Suppose that \(\displaystyle (1,\infty)\) is the set of upper bounds for \(\displaystyle C\).

That means that \(\displaystyle 1\) is not an upper bound for \(\displaystyle C\), why?

1 isn't an upper bound because it is not included in the set of upper bounds (set of upper bound is not closed)

That means \(\displaystyle \exists x\in C\) such that \(\displaystyle x>1\). WHY?

Because 1 isn't an upper bound of C there has to be an element of C such that x > 1

What is wrong with that?

I'm guessing that if there exists an x in C such that x > 1, 1 cannot be the infimum of the set of upper bounds. So there is a contradiction as the set of upper bounds cannot equal (1, infinity). Is the statement false because 1 is not included in the set of upper bounds? Does that mean the set of upper bounds must be in the form of [n, infinity) where n is a real number and the set is closed from the left.

 

[pka]

1 isn't an upper bound because it is not included in the set of upper bounds (set of upper bound is not closed)
Because 1 isn't an upper bound of C there has to be an element of C such that x > 1
I'm guessing that if there exists an x in C such that x > 1, 1 cannot be the infimum of the set of upper bounds. So there is a contradiction as the set of upper bounds cannot equal (1, infinity). Is the statement false because 1 is not included in the set of upper bounds? Does that mean the set of upper bounds must be in the form of [n, infinity) where n is a real number and the set is closed from the left.

You are correct.