Proof that limit exists and can be estimated
see attached... Let f(x) be a function defined for x>=1, f(1)=1 and df/dx=1/(x^2+f(x)^2) Prove that limf(x) exists and is less than 1+(pi/4)
see attached... Let g(a) be the real solution to x+x^5=a
Greatest Possible Value of Partial Sum
If equation A = 1 what is the greatest possible value of equation B? *(Please see attachment for equations)
Let A be a nonempty set of real numbers which is bounded below. Let -A be the set of all numbers -x where x E A. Prove that inf A = -sup(-A) (Please see the attached file for the fully formatted problem.) Included in the attachment is a copy of the solution, but please explain in your own words how the proof works; don't j ...continues
Limit Points of a Bounded Set of Real Numbers
Question: Construct a bounded set of real numbers with exactly three limit points (put the limit points at 0, 1 and 2005). (Please explain in your own words how the proof works. If you use a theorem, please state what it is and if possible, where you got it).
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 ...continues
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 ...continues
Real Analysis : Lebesgue Measure and DeMorgan's Law
Show that the sum and product of two simple functions are simple. Show that [Definition of simple: A real-valued function is called simple if it is measurable and assumes only a finite number of values. If is simple and ahs the values then , where .] This problem is from Royden's Real Analysis text for gradu ...continues
Real Analysis : Lebesgue Integral Problem
Let f be a nonnegative measurable function. Show that (integral f = 0) implies f = 0 a.e. See attached document for notations. Please help: This problem is from Royden's Chap 4 text on Lebesgue Integral.
Real Analysis : Lebesgue Integral and Monotone Convergence Theorem
Let f be a nonnegative integrable function. Show that the function F defined by F(x)= Integral[from -inf to x of f] is continuous by using the Monotone Convergence Theorem. From Royden's Real Analysis Text, chapter 4.
