# Complex Analysis Problems

## Question

1.3.4 Let S be a set consisting of two elements labeled as A and B. Deﬁne A+A = A, B+B = A, A+B = B+A = B,
A · A = A, A · B = B · A = A, and B · B = B. Show that all nine of the axioms of a ﬁeld hold for this structure

1.4.2 Show for every n ∈ N that n² ≥ n.

1.6.8 Let A be a set of real numbers and let B = {x + r : x ∈ A} for some number r. Find a relation between sup A and sup B.

1.6.11 Let A and B be sets of real numbers and write C = A ∪ B. Find a relation among sup A, sup B, and sup C

1.6.19 Using the completeness axiom, show that every nonempty set E of real numbers that is bounded below has a greatest lower bound (i.e., inf E exists and is a real number).

1.6.21 The rational numbers Q satisfy the axioms for an ordered ﬁeld. Show that the completeness axiom would not be satisﬁed. That is show that this statement is false: Every nonempty set E of rational numbers that is bounded above has a least upper bound (i.e., sup E exists and is a rational number).

1.7.2 Suppose that it is true that for each x > 0 there is an n ∈ IN so that 1/n < x. Prove the Archimedean theorem using this assumption.

1.8.1 Show that any bounded, nonempty set of natural numbers has a maximal element.

A.8.4 Prove by induction that for every n = 1,2,3, . . . (1 + x), n ≥ 1 + nx for any x > 0.

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