Introduction to Analysis

Introduction to Analysis

Homework 10

1. (Rudin ex6.18) Let γ₁, γ₂, γ₃ be curves in the complex plane, defined on [0, 2π] by γ1(t) = e^it, , γ2(t) = e^2it, γ3(t) = e2πit sin(1/t)

.

Show that these three curves have the same range, that γ₁ and γ₂ are rectifiable, that the length of γ₁ is 2π, that the length of γ₂ is 4π, and that γ₃ is not rectifiable.

2. (Rudin ex6.19) Let γ₁be a curve in R^k, defined on [a, b]; let φ be a continuous 1-1 mapping of [c, d] onto [a, b], such that φ(c) = a; and define γ₂(s) = γ₁(φ(s)). Prove that γ₂ is an arc, a closed curve, or a rectifiable curve if and only if the same is true of γ₁. Prove that γ₂ and γ₂ have the same length.

3. Let α(x) = sin x for 0 ≤ x ≤ π/2. Find the value ofRπ/2 0 xdα.

4. Let [x] denote the largest integer less than or equal to the number x. Find the value of R3

0(x²+ 1)d([x]).

5. Show that the sequence fn(x) = xⁿconverges for each x ∈ [0, 1] but that the convergence is not uniform.

6. (Rudin ex7.3)

(a) Suppose {fn} and {gn} are bounded sequences ech of which converges uniformly on a set E in a metric space X to functions f and g, respectively, Show that the sequence {f_ng_n} converges uniformly to f g on E.

(b) Construct sequences {f_n}, {g_n} which converge uniformly on some set E, but such that {f_ng_n} does not converge uniformly on E (of course, {f_ng_n} must converge on E).

7. (Rudin 7.6) Prove that the series

∞

X

n=1

(−1)ⁿx²+ n n²

converges uniformly in every bounded interval, but does not converge absolutely for any value of x.

8. (Rudin ex7.9) Let {f_n} be a sequence of continuous functions which converges uniformly to a function f on a set E. Prove that

limn→ f_n(x_n) = f (x)

for every sequence of points xn∈ E such that xn→ x, and x ∈ E. Is the converse of this true?