Without worrying about formal proofs for the moment, decide if the following sta

Question

Without worrying about formal proofs for the moment, decide if the following statements about suprema and infima are true or false. For those that are false, supply an example where the claim in question does not appear to hold.
a) A finite, nonempty set always contains its supremum.
b) If a < L for every element a in the set A, then sup A < L
c) If A and B are sets with the property that a < b for every a in A and every b in B, then it follows that sup A < inf B.
d) If sup A = s and sup B = t, then sup A + B = s + t. The set A + B is defined as A+B = {a+b : a in A and b in B}.
e) If sup A <= sup B, then there exists an element b in B that is an upperbound for A.

I think the first three are true, I don't know about D or E, but I think E is false and there does not have to be an element b in B that is an upperbound for A.

Explanation / Answer

Without worrying about formal proofs for the moment, decide if the following sta

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