Bounded Numbers (Solution in Words as well as Numbers)

Question: Let A be a nonempty set of real numbers which is bounded below. Let -A be the set of all numbers -x ... Prove that inf A = -sup(-A) (PLEASE SEE ATTACHED FOR COMPLETE PROBLEM) Included in the attachment is a copy of the solution, but please explain in your own words how the proof works; don't just copy the solution out. Please use words to describe the proof. If you use a theorem, please state what it is and if possible, where you got it.

Limits of 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: Consider c: = infα cα and d: = supα dα and show that I = ( c, d ).

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 should be in the form: "A set E in X is NOT closed if ... " * Open set. Your answer should be in the form: "A set E in X is NOT open if ... "

Real Analysis: Royden's Problem Lebesgue Measure

This problem is from Royden's Real Analysis text for graduate students "Show that the sum and product of two simple functions are simple ... " See attached document for notations.