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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