数学代写|实分析作业代写Real analysis代考|МATH1001

数学代写|实分析作业代写Real analysis代考|Limit supremum and limit infimum

In this section, we introduce concepts that help us gain a deeper understanding of the behavior of sequences. We begin with bounded sets.Upper bounds and lower bounds are not unique; for example, consider the halfopen interval $A=[-1,1)$ of real numbers. Then any number $u \geq 1$ is an upper bound of $A$ since every number in $[-1,1)$ is at most 1 . In particular, 1 is the least upper bound since no upper bound of $[-1,1)$ can be smaller than 1 without being less than some number in $[-1,1)$. For instance, $0.999$ is not an upper bound because it is less than, e.g., $0.9991$, which is in $[-1,1)$. We conclude that $\sup [-1,1)=1$

Similarly, every number $l \leq-1$ is a lower bound for $A$ and inf $[-1,1)=-1$. In this example, we also see that the supremum or the infimum may or may not be in $A$ (when they exist). Indeed, $[-1,1)$ contains not a single one of its upper bounds! Another example is that the countable set $S={1,1 / 2,1 / 3,1 / 4, \ldots}$ has sup $S=1$, which is in $S$, and $\inf S=0$, which is not in $S$.

Some sets don’t have upper bounds or lower bounds. If $A$ is $\mathbb{N}$ or $[1, \infty)$, then $\inf A=1$; lower bounds for $A$ include 0 and all negative numbers. But $A$ doesn’t have upper bounds because there is no real number $u$ that is larger than all numbers in $\mathbb{N}$ or $[1, \infty)$. In particular, sup $A$ doesn’t exist. Going further, the set $\mathbb{Z}$ of integers has no upper bounds and no lower bounds.

In Chapter 4, we discover that every bounded subset of the real number set $\mathbb{R}$ has both a supremum and an infimum; this is a consequence of the completeness of $\mathbb{R}$. But bounded sets in $\mathbb{Q}$, which is not complete, may fail to have a supremum or an infimum due to the existence of gaps or holes; see Exercise 87.
However, whencrer sup $\Lambda$ or $\inf \Lambda$ exist, they are uniquc.
Lemma 62 (a) If A has a supremum or least upper bound, then it is unique. Also, a greatest lower bound is unique, if it exists.
(b) There is a sequence $a_n$ in $A$ that converges to sup $A$. Also there is a sequence in $A$ that converges to $\inf A$.

数学代写|实分析作业代写Real analysis代考|The Real Numbers

The real numbers came up often in previous chapters where we thought of them intuitively. Now it is time to answer questions like what exactly are real numbers, and why do we need them?

We define a rational number simply as the quotient of two integers, as long as we don’t divide by 0 . This feature endows the rational numbers with a useful feature that the integers lack: the reciprocal of every rational number is also a rational number. However, this is not enough to include numbers like $\sqrt{2}$ or $\pi$ that are also extremely important in all areas of pure and applied mathematics.

The key property that defines and distinguishes real numbers from the rational ones is “completeness.” This property is different from all the algebraic properties that are common to both rational and real numbers. Completeness is a collective set property, like closure: the sum and product of two rational numbers is rational, so the set of all rational numbers is closed under addition and multiplication.
In the case of completeness, it is necessary to invoke infinity since the set of real numbers must contain the limits of so-called Cauchy infinite sequences. Loosely speaking, if we think of the decimal expansion of, say, $\pi=3.14159 \ldots$, then we can imagine it being constructed progressively using a sequence of rational numbers like 3, 3.1 $=31 / 10,3.14=314 / 100$, and so on. The proper description of this process leads to greater technical challenges, like how to define the rational numbers that make up the sequence, how to make sure that adding or multiplying (irrational) real numbers gives a real number again, and of course, how to ensure that limits of Cauchy sequences of real numbers are again real numbers, i.e., the set of all real numbers is closed under limits.

It is common practice in analysis textbooks to introduce completeness as an axiom that is added to the field axioms that define the rational numbers once the concept of Cauchy sequence has been defined. Logically equivalent versions of completeness, such as the often used “least upper bound axiom,” don’t even need Cauchy sequences and can be defined as a property that subsets of the set of real numbers must satisfy. Many textbooks introduce more than one version and then prove the equivalence of those versions.

This approach certainly makes it possible to enter the theory faster, but it offers little intuition and insufficient insight into the real numbers, or in particular, what the irrational numbers are. From a pedagogical point of view, the axiomatic approach to introducing completeness seems contrived.

实分析代考

数学代写|实分析作业代写Real analysis代考|Limit supremum and limit infimum

(b) 有一个序列一个n在一个收敛于 sup一个. 也有一个顺序一个收敛于信息一个.

数学代写|实分析作业代写Real analysis代考|The Real Numbers

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