Prove that if A and B are two sups for set S that A = B
Prove that if A and B are two sups for set S that A = B.
There are two equivalent definitions of sup(S).
Definition 1: M = sup(S) iff
a. For all sεS, M ≥ s
b. If for all sεS, K ≥ s, then K≥M
Definition 2: M = sup(S) iff
a. For all sεS, M ≥ s
b. If K K
First let’s prove the definitions are equivalent.
Suppose N = sup(S) in the sense of definition 1 and let K B. Then there exists sεS, s > K. But this contradicts B is an upper bound for S. Similarly if B > A, we get a contradiction. Therefore A=B.
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