# 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, \$\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\$.