Open and Closed Sets, Differentiability and Sequences
1. Let A and B be two nonempty sets of real numbers. Define A+B = {a+b: a Є A and b Є B}. (a) Show that if A is open, then A+B is open. (b) If A and B are both closed, is A+B closed? Justify your answer. 2. Let f be differentiable for x>a and f(x)/x --> A as x-->∞ . Prove that there is a sequence xn--> ...continues
3. Let f be differentiable for x>a and f(x) + f'(x)-->A as x--> ∞ . Prove that f(x)--> A and f'(x)-->0. 4. If A and B are nonempty subsets of R that are bounded from above, prove that the sup(A+B) = sup(A)+sup(B)
Countability, Monotonic Sequences, Convergence, Continuity and One-to-One
5. Let the points of any countable subset E of (a,b), which may be dense, be arranged in a sequence { }. Let { } be a sequence of positive numbers such that converges. Define ( ) (i.e., sum over those indices n for which ). Verify the following properties of : (a) monotonically increasing on (a,b); (b) discontinuous at ev ...continues
7. If ∑an converges and if {bn} is monotone and bounded, prove that ∑anbn converges. 8. Prove that f(x)= x^1/2 is uniformly continuous on [0,∞). ---
Let E C R1 and let D be a dense subset of E. If are continuous real-valued functions on E for n=1,2,…, and fn converges uniformly on D, prove that fn converges uniformly on E. (See attached file for full problem description) I am using the book Methods of Real analysis by Richard Goldberg.
2. Denote by -P the set of all negative integers, i.e., the set to which the number m belongs only in case there is a member n of P such that m = -n. If the number m is in -P and Z is a number such that m
Stone-Weierstrass Approximation Theorem
Show that there is a countable set F of functions of form x--->a_0 + a_1cosx + a_2cos(2x) + ... +a_ncos(nx) such that every continuous real-valued function on [0,pi] is the uniform limit of a sequence of functions (f_n) in F.
Applications of Mean Value Theorem : Real and Simple Roots of Polynomials
Show that if the roots of the polynomial p are all real, then the roots of p' are all real. If, in addition, the roots of p are all simple, then the roots of p' are all simple.
Discuss the differentiability of each of the following functions at all real numbers and find its derivative at those real numbers at which it is differentiable. See attached file for full problem description.
Convergence or Divergence of Integrals
The problems in the file submitted are from an undergraduate course in real Analysis. If you are able to work the problems, please detail any theorems or lemmas used in your solutions. The book we are using is titled "The Elements of Real Analysis" by Robert G. Bartle. We are working on derivatives and integrals, but have not ...continues
