LUB, interval and pigeonhole proofs
1. Let S be a subset of R. Show that S is an inverval if and only if for all a,b in S, we have [a,b] contained in S.
2. two parts:
prove the following for A,B being bounded sets of the real numbers.
part a: if A+B is defined as the set of all a + b such that a is in A and b is in B
LUB(A + B) = LUB(A) + LUB(B)
and the second part:
part b: if every element in A,B is positive, and
A*B is defined as the set of all a*b such that a is in A and b is in B
LUB(A*B) = LUB(A) * LUB(B)
3. (this one is hard!) (supposed to be done by Pigeon hole)
Let T be a triangle such that T can be placed inside the unit square (sides of length one) with the center of the square not contained in the triangle. Prove that one of the sides of T must have a length of less than 1.
Any ideas? I appreciate the help!