# Thread: more help upper bounds

1. ## more help upper bounds

1) Let A and B be a subsets of the real numbers with least upper bound u and v. Prove that their union has a least upper bound, and express it in terms of u and v.

2) Let A be the set of negative real numbers. Prove that 0 is equal to the least upper bound of A.

Hint: one needs to check that 0 is an upper bound and if x < 0 then 0 is not an upper bound; i.e., there is some y in A such that x < y.

I am soo sorry to keep on asking questions, but Math is reallly hard for me. Thank you very much in advance.

2. Originally Posted by jenjen
2) Let A be the set of negative real numbers. Prove that 0 is equal to the least upper bound of A.
Assume, $\displaystyle A$ has a least upper bound $\displaystyle x$.

By trichtonomy,
$\displaystyle x<0$
$\displaystyle x=0$
$\displaystyle x>0$

If, $\displaystyle x<0$
Then, $\displaystyle x/2<0$ element of $\displaystyle A$
And, $\displaystyle x/2>x$

If, $\displaystyle x>0$
Then, $\displaystyle x/2>0$ is an upper bound.
And, $\displaystyle x/2<x$.
Thus, $\displaystyle x=0$ if the least upper bound exists.
Use the completeness property. $\displaystyle A,B$ are upper bounded. Then, $\displaystyle A\cup B$ is upper bounded. Thus, by completeness it has a least upper bound.