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1. Let A and B be subsets of R, each of which A and B be...

1. Let A and B be subsets of R, each of which A and B be subsets of R, each of which has a minimum element. Prove that if A ⊆ B, then min A ≥ min B.

2.. Let a and b be real numbers such that a < b. Prove that a < a + b / 2 < b. This number a + b / 2 is called the arithmetic mean of a and b.

3.. Let a, b and c be real numbers such that a < b and a < c. Prove that b < c if and only if [a, b) ⊂ [a, c)..

4. Let A ⊆ R, c ∈ R, and suppose A has a maximum element. Prove that c is an upper bound for A if and only if c ≥ max A.

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