Recent content by The Captain

Henstock-Kurzweil Integral is used in Real Analysis. I'm pretty sure my class is being taught material that is generally left for graduate school, however since my professor did his PhD studies on it, thinks we can handle it. As for the second equation, we are to think of f(x,y) as a...

Homework Statement Prove \int\int_{[-1,1]×[-1,1]}f(x,y)dA is not Henstock Integrable. Homework Equations f(x,y) = \frac{xy}{(x^{2}+y^{2})^{2}} f(x,y) = 0 if x^{2}+y^{2}=0 on the region [-1,1]×[-1,1] The Attempt at a Solution The only hints given is that we will not be able to solve...

Intersection of a sequence of intervals equals a point (Analysis) Homework Statement Let A_{n} = [a_{n}, b_{n}] be a sequence of intervals s.t. A_{n}>A_{n+1} and |b_{n}-a_{n}|\rightarrow0. Then \cap^{∞}_{n=1}A_{n}={p} for some p\inR. Homework Equations Monotonic Convergent Theorem If...

What if I rewrote the last part as: Since A \subseteq B , \forall x \in B and \forall y \in A then y \leq x so a \leq x \leq b , then a \leq b . Therefore sup(A) \leq sup(B) .

Assume A = \left\{ 1- \frac{1}{n} : \ n \in \mathbb Z^{+} \right\} , prove sup(A) = 1. If 1 is the least upperbound such that \forall \epsilon > 0, 1 - \epsilon is not an upperbound of A, then \exist a \in A: \ a \in \left[ 1 - \epsilon , 1 \right) , then a = 1 - \frac{1}{n_{0}} for...

Could I word it this way? Suppose \exists b \in B : \ b=sup(B), \ \forall x \in B : x \leq b . Then \exists a \in A: \ a=sup(A), \forall y \in A: \ y \leq a. \exists m \in \mathbb R : m \geq b \ \forall b \in B, B \subset \left( - \infty , m \right] Since A \subseteq B, \ \forall x \in A...

Suppose b=sup(B), \forall x \in B : x \leq b and a=sup(A), \forall y \in A : y \leq a . \exists m \in \mathbb R : m \geq b \ \forall b \in B, B \subset \left( - \infty , m \right] Since A \subseteq B, \ \forall x \in A , then y \leq a \leq x \leq b , and since a=sup(A) and...

Without proving it, just explaining it: If some element of A is in B, and the sup(B) is the least upper bound of B, then sup(A) is less than or equal to the sup(B).

So what you're saying is that I'm assuming there is an element in B, such that it is less than or equal to the sup(B), and then taking another element from A and because A is a subset of B, that the element in A is less than or equal to the sup(B)?

Didn't think that it would have been that easy.

Proving sup(a) \leq sup(B)

Homework Statement Given A and B are sets of numbers, A \neq \left\{ \right\} , B is bounded above, and A \subseteq B . Explain why sup(A) and sup(B) exist and why sup(A) \leq sup(B).Homework Equations \exists r \in \mathbb R \: : \: r \geq a \: \forall a \in A \exists r \in \mathbb R \: ...

I had to prove that \mathbb Z^{+} \: X \: Z^{+} \: \rightarrow \: Z^{+} was one-one and onto using f(a,b)=2^{a-1}(2b-1), does that count for proving it's countable, and if it's not, no I don't know how to prove it's countable. The class I'm taking is a giant leap from Calc 4, and Abstract...

Homework Statement Prove that \mathbb Z^{+} X \: \mathbb Z^{+} X \: \mathbb Z^{+} is countable, where X is the Cartesian product.Homework Equations The Attempt at a Solution I'm lost as to where to start proving this.

So I need to prove that if f(y)=Y and f^{1}(Y)=y, that f(f^{1}(Y))=Y?