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...
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...
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.
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)...