HW1 An abelian group is a set S together with a binary operation + so that (1) x + y = y + x for any x, y ∈ S (2) (x + y

1. HW1

An abelian group is a set S together with a binary operation + so that (1) x + y = y + x for any x, y ∈ S

(2) (x + y) + z = x + (y + z),

(3) for each x ∈ S, there is 0 in S so that x + 0 = 0 + x = x

(4) for each x ∈ S, there is −x in S so that x + (−x) = (−x) + x = 0.

A ring is a nonempty set R together with two binary operations +, · such that (1) (R, +) forms an abelian group,

(2) (a · b) · c = a · (b · c) for all a, b, c ∈ R, (3) a · (b + c) = a · b + a · c.

If in addition, ab = ba for all a, b ∈ R, then R is said to be a commutative ring. If R contains an element 1R such that a · 1R = 1R· a for all a ∈ R, then R is said to be a ring with identity. A field is a commutative ring (R, +, ·) with identity so that (R \ {0}, ·) is an abelian group.

1. Let R² be the set of ordered pairs (x1, x2) with x1, x2 ∈ R. Define the addition on R² by

(x₁, x₂) + (y₁, y₂) = (x₁+ x₂, y₁+ y₂), and the multiplication by

(x1, x2) · (y1, y2) = (x1y1− x₂y2, x1y2+ x2y1).

Here (x₁, x₂), (y₁, y₂) ∈ R².

(a) Show that C = (R², +, ·) forms a commutative ring with 1_C, where 1_C= (1, 0).

(b) Assume that (x1, x2) 6= (0, 0). Find (y1, y2) so that (x₁, x₂) · (y₁, y₂) = 1_C. This implies that C is a field.

A metric space consists of a pair (M, d), where M is a nonempty set and d : M × M → [0, ∞)

is a function, called the distance function or the metric, such that for any x, y, z ∈ M, (1) d(x, y) = d(y, x)

(2) d(x, y) = 0 if and only if x = y, (3) d(x, z) ≤ d(x, y) + d(y, z).

2. Let z = a + bi be a complex number. The absolute value of z is defined to be

|z| = √

a²+ b². Let d_C(z, w) = |z − w| for z, w ∈ C. Show that (C, dC) is a metric space.

A sequence of complex numbers is a function z : N → C. Denote z(n) by zn and z by (z_n).

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3. A sequence (zn) of complex numbers is bounded if there exists M > 0 so that

|z_n| ≤ M for all n ≥ 1. Show that a convergent sequence is always bounded.

5. Let f : N → N be a function. We say that f is increasing if f (i) < f (j) for any i < j.

(a) Let f : N → N be an increasing function. Show that f (i) ≥ i for all i ≥ 1.

(b) Let z = (zn) be a sequence of complex numbers. We say that a sequence w = (w_n) is a subsequence of z = (z_n) if there exists an increasing function f : N → N so that w = z ◦ f, i.e. wn = z_{f (n)} for all n ≥ 1. Let (zn) be a convergent sequence in C whose limit is z. Show that all the subsequences of (z_n) converge to z.

6. Let z, w ∈ C. Show that

z − w 1 − zw

< 1 if |z| < 1 and |w| < 1.