Real Analysis I
Prove the following exercises (show all your work)-
Exercise 1.1.1: Prove part (iii) of Proposition 1.1.8. That is, let F be an ordered field and x, y,z ∈ F. Prove If x < 0 and y < z, then xy > xz.
Let F be an ordered field and x, y,z,w ∈ F. Then:
If x < 0 and y < z, then xy > xz.
Exercise 1.1.5: Let S be an ordered set. Let A ⊂ S and suppose b is an upper bound for A. Suppose b ∈ A. Show that b = sup A.
Exercise 1.1.6: Let S be an ordered set. Let A ⊂ S be a nonempty subset that is bounded above. Suppose sup A exists and sup A ∈/ A. Show that A contains a countably infinite subset.
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