# more help upper bounds

• Nov 14th 2006, 07:34 PM
jenjen
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.
• Nov 14th 2006, 07:48 PM
ThePerfectHacker
Quote:

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, $A$ has a least upper bound $x$.

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

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

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