# Thread: Suprema question

1. ## Suprema question

Let S,T be non-empty bounded subsets of the real numbers. Prove that $\displaystyle S \cup T$ is bounded above.
How would you start this question?

2. Originally Posted by worc3247
Let S,T be non-empty bounded subsets of the real numbers. Prove that $\displaystyle S \cup T$ is bounded above.
How would you start this question?

If $\displaystyle s\geq 0,\;t\geq 0$ are upper bounds for $\displaystyle S$ and $\displaystyle T$ respectively, prove that $\displaystyle x\leq s+t$ for all $\displaystyle x\in S\cup T$ .

Fernando Revilla

3. Thats not right, you should take max{s,t} as upper bound. For example think of S=T=[-2,-1] and s=t=-1. In this case s+t=-2 is no upper bound of the union.

4. Originally Posted by Kevin123
Thats not right. mFor example think of S=T=[-2,-1] and s=t=-1. In this case s+t=-2 is no upper bound of the union.[/
It is right, I said

$\displaystyle s\geq 0,\;t\geq 0$

upper bounds, not minimum upper bounds.

you should take max{s,t}
Or

$\displaystyle \max \{s,t\}+\pi$

Fernando Revilla

5. Oh sorry, I did not see that you assume s,t >= 0. Of course then your answer is right... :-)

6. I think I can see the general idea, but am having trouble writing a formal proof. Here is what I have thus far:
Let $\displaystyle \alpha \geq 0$and $\displaystyle \beta \geq 0$ such that $\displaystyle \alpha , \beta$ are upper bounds for S and T respectively.
If we set $\displaystyle l =$max{$\displaystyle \alpha , \beta$}, then:
$\displaystyle l \geq x, \forall x \in S \cup T$. Therefore $\displaystyle S \cup T$ is bounded above.

Is this along the right lines or am I way off?

7. You cannot assume that the upper bound is in S. Think of S=(0,1). Then any upper bound is greater or equal 1.

8. Ok, so I need to set $\displaystyle \alpha$ and $\displaystyle \beta$ so that they are greater than the supremum of the sets?