### 数学代写|实变函数作业代写Real analysis代考|The Supremum and Infimum of a Set of Real Numbers

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• Foundations of Data Science 数据科学基础

## 数学代写|实变函数作业代写Real analysis代考|Bounded Sets

Definition 2.6.1 Sets Bounded Above
We say a nonempty set $S$ is bounded above if there is a number $M$ so that $x \leq M$ for all $x$ in $S$. We call $M$ an upper bound of $S$ or just an u.b.

Example 2.6.1 If $S=\left{y: y=x^{2}\right.$ and $\left.-1 \leq x \leq 2\right}$, there are many u.b.’s of $S$. Some choices are $M=5, M=4.1$. Note $M=1.9$ is not an u.b. You should draw a graph of this to help you understand what is going on.

Example 2.6.2 If $S={y: y=\tanh (x)$ and $x \in \Re}$, there are many u.b.’s of S. Some choices are $M=2, M=2.1$. Note $M=0$ is not an u.b. Draw a picture of this graph too.
We can also talk about a set being bounded below.
Definition 2.6.2 Sets Bounded Below
We say a set $S$ is bounded below if there is a number $m$ so that $x \geq m$ for all $x$ in $S$. We call $m$ a lower bound of $S$ or just $a$ l.b.
Example 2.6.3 If $S=\left{y: y=x^{2}\right.$ and $\left.-1 \leq x \leq 2\right}$, there are many l.b.’s of $S$. Some choices are $m=-2, m=-0.1$. Note $m=0.3$ is not a l.b.

Example 2.6.4 If $S={y: y=\tanh (x)$ and $x \in \Re}$, there are many l.b.’s of S. Some choices are $m=-1.1, m=-1.05$. Note $m=-0.87$ is not $a$ l.b. Draw a picture of this graph again.
We can then combine these ideas into a definition of what it means for a set to be bounded.
Definition 2.6.3 Bounded Sets
We say a set $S$ is bounded if $S$ is bounded above and bounded below. That is, there are finite numbers $m$ and $M$ so that $m \leq x \leq M$ for all $x \in S$. We usually overestimate the bound even more and say $S$ is bounded if we can find a number $B$ so that $|x| \leq B$ for all $x \in S$. A good choice of such $a B$ is to let $B=\max (|m|,|M|)$ for any choice of l.b. $m$ and u.b. $M$.
Example 2.6.5 If $S=\left{y: y=x^{2}\right.$ and $\left.-1 \leq x<2\right}$, here $S=[0,4)$ and so for $m=-2$ and $M=5$, a choice of $B$ is $B=5$. Of course, there are many other choices of $B$. Another choice of $m$ is $m=-1.05$ and with $M=2.1$, we could use $B=2.1$.

Example 2.6.6 If $S={y: y=\tanh (x)$ and $x \in \Re}$, we have $S=(-1,1)$ and for $m=-1.1$ and $M=1.2$, a choice of $B$ is $B=1.2$.

## 数学代写|实变函数作业代写Real analysis代考|Least Upper Bounds and Greatest Lower Bounds

The next material is more abstract! We need to introduce the notion of least upper bound and greatest lower bound. We also call the least upper bound the l.u.b. It is also called the supremum of the set $S$. We use the notation $\sup (S)$ as well. We also call the greatest lower bound the g.l.b. It is also called the infimum of the set $S$. We use the notation $\inf (S)$ as well.
Definition 2.6.4 Least Upper Bound and Greatest Lower Bound
The least upper bound, l.u.b. or sup of the set $S$ is a number $U$ satisfying

1. $U$ is an upper bound of $S$
2. If $M$ is any other upper bound of $S$, then $U \leq M$.
The greatest lower bound, g.l.b. or inf of the set $S$ is a number u satisfying
3. $u$ is a lower bound of $S$
4. If $m$ is any other lower bound of $S$, then $u \geq m$.

Example 2.6.7 If $S=\left{y: y=x^{2}\right.$ and $\left.-1 \leq x<2\right}$, here $S=[0,4)$ and so inf $(S)=0$ and $\sup (S)=4$

Example 2.6.8 If $S={y: y=\tanh (x)$ and $x \in \Re}$, we have $\inf (S)=-1$ and $\sup (S)=1$. Note the inf and sup of a set $S$ need NOT be in $S$ !

Example 2.6.9 If $S={y: \cos (2 n \pi / 3), \quad \forall n \in \mathbb{N}}$, The only possible values in $S$ are $\cos (2 \pi / 3)=$ $-1 / 2, \cos (4 \pi / 3)=-1 / 2$, and $\cos (6 \pi / 3)=1$. There are no other values and these 2 values are endlessly repeated in a cycle. Here $\inf (S)=-1 / 2$ and $\sup (S)=1$.

Comment 2.6.1 If a set $S$ has no finite lower bound, we set $\inf (S)=-\infty$. If a set $S$ has no finite upper bound, we set $\sup (S)=\infty$.
Comment 2.6.2 If the set $S=0$, we set $\inf (S)=\infty$ and $\sup (S)=-\infty$.
These ideas then lead to the notion of the minimum and maximum of a set.
Definition 2.6.5 Maximum and Minimum of a Set
We say $Q \in S$ is a maximum of $S$ if $\sup (S)=Q$. This is the same, of course, as saying $x \leq Q$ for all $x$ in $S$ which is the usual definition of an upper bound. But this is different as $Q$ is in $S$. We call $Q$ a maximizer or a maximum element of $S$.
We say $q \in S$ is a minimum of $S$ if $\inf (S)=q$. Again, this is the same as saying $x \geq q$ for all $x$ in $S$ which is the usual definition of a lower bound. But this is different as $q$ is in $S$.
We call $q$ a minimizer or a minimal element of $S$.

## 数学代写|实变函数作业代写Real analysis代考|The Completeness Axiom and Consequences

There is a fundamental axiom about the behavior of the real numbers which is very important.
Axiom 1 The Completeness Axiom
Let $S$ be a set of real numbers which is nonempty and bounded above. Then the supremum of $S$ exists and is finite.
Let $S$ be a set of real numbers which is nonempty and bounded below. Then the infimum of $S$ exists and is finite.
Comment 2.6.3 So nonempty bounded sets of real numbers always have a finite infimum and supremum. This does not say the set has a finite minimum and finite maximum. Another way of saying this is that we don’t know if $S$ has a minimizer and maximizer.
We can prove some basic results about these things.
Theorem 2.6.1 A Set has a Maximum if and only if its Supremum is in the Set
Let $S$ be a nonempty set of real numbers which is bounded above. Then $\sup (S)$ exists and is finite. Then $S$ has a maximal element if and only if $(I F F) \sup (S) \in S$. We also use the symbol $\Longleftrightarrow$ to indicate IFF.
Proof 2.6.1
$(\Leftarrow)$ :
Assume $\sup (S)$ is in $S$. By definition, $\sup (S)$ is an upper bound of $S$ and so must satisfy $x \leq \sup (S)$ for all $x$ in $S$. This says $\sup (S)$ is a maximizer of $S$.
$(\Rightarrow)$ :
Let $Q$ denote a maximizer of $S$. Then by definition $x \leq Q$ for all $x$ in $S$ and is an upper bound. So by

the definition of a supremum, $\sup (S) \leq Q$. Since $Q$ is a maximizer, $Q$ is in $S$ and from the definition of upper bound, we have $Q \leq \sup (S)$ as well. This says $\sup (S) \leq Q \leq \sup (S)$ or $\sup (S)=Q$.
Theorem 2.6.2 A Set has a Minimum if and only if its Infimum is in the Set
Let $S$ be a nonempty set of real numbers which is bounded below. Then $\inf (S)$ exists and is finite. Then
$S$ has a minimal element $\Longleftrightarrow \inf (S) \in S$.
Proof 2.6.2
$(\Leftarrow)$ : Assume $\inf (S)$ is in $S$. By definition, $\inf (S)$ is a lower bound of $S$ and so must satisfy $x \geq$ $\inf (S)$ for all $x$ in $S$. This says $\inf (S)$ is a minimizer of $S$.
$(\Rightarrow)$ : Let $q$ denote a minimizer of $S$. Then by definition $x \geq q$ for all $x$ in $S$ and is a lower bound. So by the definition of an infimum, $q \leq \inf (S)$. Since $q$ is a minimizer, $q$ is in $S$ and from the definition of lower bound, we have $\inf (S) \leq q$ as well. This says $\inf (S) \leq q \leq \inf (S)$ or $\inf (S)=q$.

## 数学代写|实变函数作业代写Real analysis代考|Least Upper Bounds and Greatest Lower Bounds

1. 在是一个上限小号
2. 如果米是的任何其他上界小号， 然后在≤米.
集合的最大下限、glb 或 inf小号是一个你满意的数字
3. 在是的下界小号
4. 如果米是的任何其他下限小号， 然后在≥米.

## 数学代写|实变函数作业代写Real analysis代考|The Completeness Axiom and Consequences

(⇐)：

(⇒):

Let小号是一组有界的非空实数。然后信息(小号)存在并且是有限的。然后

(⇐)： 认为信息(小号)在小号. 根据定义，信息(小号)是的下界小号所以必须满足X≥ 信息(小号)对全部X在小号. 这说信息(小号)是一个极小值小号.
(⇒)： 让q表示一个最小化器小号. 然后根据定义X≥q对全部X在小号并且是一个下界。所以根据下确界的定义，q≤信息(小号). 自从q是一个最小化器，q在小号并且根据下界的定义，我们有信息(小号)≤q也是。这说信息(小号)≤q≤信息(小号)或者信息(小号)=q.

## 有限元方法代写

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## MATLAB代写

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