# Search results

1. ### Henstock Integral Help

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...
2. ### Henstock Integral Help

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...
3. ### Intersection of a sequence of intervals equals a point

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...
4. ### Sup(A) less than or equal to sup(B)

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) .
5. ### Sup(A) less than or equal to 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...
6. ### Sup(A) less than or equal to sup(B)

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...
7. ### Sup(A) less than or equal to sup(B)

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...
8. ### Sup(A) less than or equal to sup(B)

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).
9. ### Sup(A) less than or equal to 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)?
10. ### Sup(A) less than or equal to sup(B)

Didn't think that it would have been that easy.
11. ### Sup(A) less than or equal to sup(B)

Proving sup(a) \leq sup(B)
12. ### Sup(A) less than or equal to 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...
13. ### Prove is countable

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...
14. ### Prove is countable

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.
15. ### Prove onto

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

Onto means that for a function f:A \rightarrow B if \forall b \in B there is an a \in A: f(a)=b The inverse means that if you take the f^{-1}(b) that it should map back to a?
17. ### Prove onto

Homework Statement Prove that if f: X \rightarrow Y is onto, then f(f^{-1}(B))=B \forall B \in Y Homework Equations The Attempt at a Solution