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Real Analysis

Real Analysis : Fold Lines

By an n-fold line subdivision of the plane P, we mean any collection of n-distinct (infinite) lines in P, together with the open regions in P that they determine. (We don't count the lines as part of the regions.) Let us say that two such regions are adjacent if their boundaries have a positive-length or infinite line segment

Real analysis

Give formal negations of the following definitions: * Limit point. Your answer should be in the form: "A point p in X is NOT a limit point of the set E in X if ... " * Interior point. Your answer should be in the form: "A point p in X is NOT an interior point of the set E in X if ... " * Closed set. Your answer

Real Analysis - Open Intervals

Fix a point p in R. Let { Iα } be a ( possibly infinite ) collection of open intervals Iα = ( cα , dα ) which is a subset of R, such that pЄ Iα for all α. Prove that the union I: = Uα Iα is also an open interval ( possibly infinite ). Hint: Cons


Find the limit(as R goes to infinity) of (-1/2+(1/4+V/f*R)^1/2)/(V/f*R)

Real Analysis...Open Sets

Prove that the open interval...with a, b being real numbers is an open set. (See attachment for full question)


Find the limit lim 3x/x+2 x->-2+


Lim sqrt (x+3) - sqrt (3)/x x->o

Rootfinding for Nonlinear Equations

24. Which of the following iterations will converge to the indicated fixed point alpha (provided x_0 is sufficiently close to alpha)? If it does converge, five the order of convergence, for linear convergence, give the rate of linear convergence. a) x_n+1 = -16 + 6x_m + 12/x_n alpha = 2 b) x_n+1 = 2/3x_n + 1/(x_n)^2

Using a Summation Series to Estimate a Quantity

Say the only tool given to you is a calculator which performs addition, subtraction, multiplication, and division. Let X= Summation (k=1 -->n) e^-(k/n)^2 with N^20 Explain a practical way of computing X within an error of 10^8. Roughly how big is X?

Covers and Convergent Series

(4) (a) Let I1,I2,I3... be open intervals and let J be a closed interval and let J be a closed inteval. Let lk be the length of Ik, and let L be the length of J....Please see the attachement

Real Analysis : Subintervals

Prove rigorously: Let N be an integer > or equal to 2, and let Xsub0....Xsubn E [0,1). Prove that there exist i and j with i not equal to j such that abs (xsubi-xsubj) < 1/n.

Real Analysis : Bounded Continuity / Differentiability

Problem: Let f: [0, &#8734;) &#8594; R be a bounded function. For all X greater than or equal to 0, let G(x)=sup{f(t): 0 is less than or equal to t is less than or equal to x} a) Show that if f is continuous, g is also continuous. Is the converse also true? Justify. b) If f is differentiable and continuous, is g also d

Real Analysis

29.18 Let f be a differentiable on R with a = sup {|f &#8242;(x)|: x in R} < 1. Select s0 in R and define sn = f (sn-1) for n &#8805; 1. Thus s1 = f (s0), s2 = f(s1), etc Prove that (sn) is a convergence sequence. Hint: To show (sn) is Cauchy, first show that |sn+1 - sn| &#8804; a&#1468;|sn - sn-1| for n &#8805; 1.

Real Analysis

29.12 (a) Show that x < tan x for all x in (0, &#960;/2). (b) Show that x/ sin x is strictly increasing function on (0, &#960;/2). (c) Show that x &#8804; (&#960;/2)&#1468;sin x for all x in [0, &#960;/2].

Real Analysis

29.2 Prove that |cos x - cos y| &#8804; |x - y| for all x, y in R

Real Analysis : Constant Function

Let f be defined on R and suppose it satisfies |f(x + y) - f(x)| &#8804; |y|^(3/2) for all x, yER. Show that f is a constant function.