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# 数学代写|微积分代考calculus assignment|PROOFS

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## 数学代写|微积分代考calculus assignment|The nested interval property

There is some unapologetic use of compact intervals in a way that might not satisfy the most rigorous requirements. We construct them when we need them (without using sups and infs) and we assume that we can use this important property:
Nested interval property: If $\left[a_{1}, b_{1}\right] \supset\left[a_{2}, b_{2}\right] \supset\left[a_{3}, b_{3}\right] \ldots$ is a shrinking sequence of compact intervals with lengths decreasing to zero,
$$\lim {n \rightarrow \infty}\left(b{n}-a_{n}\right)=0$$
then there is a unique point $z$ that belongs to each of the intervals.
There are only two instances where the property is used but they are crucial. The first is for establishing the Cousin covering lemma; the second for giving the Cauchy criterion for integrability of a function.

## 数学代写|微积分代考calculus assignment|Managing epsilons

The student should have had some familiarity with $\epsilon, \delta$ proofs. Certainly the notions of continuity, limits, and derivatives in an earlier course will have prepared the rudiments.

In such arguments here we frequently have several steps or many steps. With two steps the student is by now accustomed to splitting $\epsilon$ into two pieces
$$\epsilon=\frac{\epsilon}{2}+\frac{\epsilon}{2} .$$
Then the argument needed to show something is smaller than $\epsilon$ breaks into showing the two separate pieces are smaller than $\frac{\epsilon}{2}$.

It is only moderately more difficulty to handle infinitely many pieces in a proof. In the calculus as presented in these notes we frequently have to handle some condition described by an infinite sequence of steps. For that a very simple device is available, similar to splitting the $\epsilon$ into two or three or more pieces:
$$\epsilon=\frac{\epsilon}{2}+\frac{\epsilon}{4}+\frac{\epsilon}{8}+\frac{\epsilon}{16}+\cdots+\frac{\epsilon}{2^{n}}+\ldots$$

## 数学代写|微积分代考CALCULUS ASSIGNMENT|Full covering arguments

While there may appear to be a lot of material presented in the notes (Chapters 1-4), by far the bulk of it uses one basic kind of argument, relying only on the manipulation of full covers. This one technique, once mastered, will carry the student through nearly all of the proofs and exercises.

Indeed the course itself could be defined as going only to the extremes that can be handled by full covering arguments. In Chapter 4 we are just on the edge of introducing measure theory but hold back by relying only on full coverings. Thus sets of measure zero (null sets) are defined and handled only by full covers. Similarly the growth of a continuous function on a set could easily lead to the study of variational measures, but reduces instead to a simple full cover statement.

There is a dual notion: a fine covering argument, i.e., what most would call a Vitali covering argument. That is the line that the course does not cross until Chapter 5 , and that line defines what we can prove and what we cannot prove until the advanced material.

## 数学代写|微积分代考CALCULUS ASSIGNMENT|THE NESTED INTERVAL PROPERTY

$$\lim {n \rightarrow \infty}\left(b{n}-a_{n}\right)=0$$

## 数学代写|微积分代考CALCULUS ASSIGNMENT|MANAGING EPSILONS

ε=ε2+ε2.

ε=ε2+ε4+ε8+ε16+⋯+ε2n+…

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