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)?
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.
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?
Homework Statement
Suppose A={\frac{1}{1},\frac{1}{2},....}={\frac{1}{n}|n\in{Z^+}}
Homework Equations
The Attempt at a Solution
Could you take the limit of \frac{1}{n} as \infty to prove this, or would I go about it a different route?
Homework Statement
Define f: Z+ X Z+ -> Z+ by
f(a,b) = 2^(a-1)(2b-1) for all a,b in Z+
where Z+ is the set of all positive integers,
and X is the Cartesian product
Homework Equations
The Attempt at a Solution
If we assume (a,b) as ordered pairs and write them as follows:
(1,1) (1,2)...