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# 数学代写|微积分代写Calculus代考|MATH1023 Sets, sequences, functions

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## 数学代写|微积分代写Calculus代考|Sets, sequences, functions

2.1 Theoretical background
In this section, we recall the definitions of a few notions that will be used in the sequel.
2.1.1 Upper and lower bound of a set
Definition 2.1 Let $\mathcal{A}$ be a subset of $\mathbb{R}$. A real number $L$ is an upper bound of $\mathcal{A}$ if and only if $x \leq L$ for all $x \in \mathcal{A}$.

Definition 2.2 Let $\mathcal{A}$ be a subset of $\mathbb{R}$. A real number $l$ is a lower bound of $\mathcal{A}$ if and only if $x \geq l$ for all $x \in \mathcal{A}$.
2.1.2 Maximum and minimum of a set
Definition 2.3 Let $\mathcal{A}$ be a subset of $\mathbb{R}$. A real number $M$ is the maximum of $\mathcal{A}$ if and only if $M$ is an upper bound of $\mathcal{A}$ and $M \in \mathcal{A}$.

Definition 2.4 Let $\mathcal{A}$ be a subset of $\mathbb{R}$. A real number $m$ is the minimum of $A$ if and only if $m$ is a lower bound of $\mathcal{A}$ and $m \in \mathcal{A}$.
2.1.3 Supremum and infimum of a set
We recall two equivalent definitions of supremum of a set of real numbers.

Definition 2.5 The supremum of a subset $\mathcal{A}$ of $\mathbb{R}$ is the least upper bound.
Definition 2.6 A real number $L$ is the supremum of a subset $\mathcal{A}$ of $\mathbb{R}$ if and only if the following two conditions hold:
a) $x \leq L$ for all $x \in \mathcal{A}$ (i.e., $L$ is an upper bound);
b) for all $\epsilon>0$ there exists $x \in \mathcal{A}$ such that $x>L-\epsilon$ (i.e., $L$ is the least upper bound).

Also for the infimum of a set of real numbers we have two equivalent definitions:
Definition 2.7 The infimum of a subset $\mathcal{A}$ of $\mathbb{R}$ is the greatest lower bound.
Definition 2.8 A real number $l$ is the infimum of a subset $\mathcal{A}$ of $\mathbb{R}$ if and only if the following two conditions hold:
a) $x \geq l$ for all $x \in \mathcal{A}$ (i.e., $l$ is a lower bound);
b) for all $\epsilon>0$ there exists $x \in \mathcal{A}$ such that $x<l+\epsilon$ (i.e., $l$ is the greatest lower bound).
2.1.4 Limit point of a set
We recall two equivalent definitions of limit point.
Definition 2.9 The real number $c$ is a limit point of a subset $\mathcal{A}$ of $\mathbb{R}$ if and only if any neighbourhood of $c$ contains an infinite number of elements of $\mathcal{A}$.

Definition 2.10 The real number $c$ is a limit point of a subset $\mathcal{A}$ of $\mathbb{R}$ if and only if any neighbourhood of $c$ contains at least an element of $\mathcal{A}$ different from $c$.

Remark 2.11 In the definition of limit point, the condition $c \in \mathcal{A}$ is not required. Thus, there are cases where the limit point $c$ belongs to $\mathcal{A}$, and cases where it does not belong to $\mathcal{A}$.

## 数学代写|微积分代写Calculus代考|Domain, codomain and image of a function

Given two subsets $\mathcal{A}$ and $\mathcal{B}$ of $\mathbb{R}$, and a function $f$ from $\mathcal{A}$ to $\mathcal{B}$, the set $\mathcal{A}$ is called domain of $f$ and the set $\mathcal{B}$ is called codomain of $f$. The codomain is the set of destination of $f$ and should not be confused with the image of $f$, defined by
$$\operatorname{Im}(f)={y \in \mathcal{B}: \text { there exists } x \in \mathcal{A} \text { such that } f(x)=y} .$$
The domain of a function $f$ is also denoted by $D(f)$.

Remark 2.12 For real-valued functions, as those discussed in this book, if the codomain $\mathcal{B}$ is not explicitly indicated, then it is understood that $\mathcal{B}=\mathbb{R}$. We note that, for functions defined by a formula, unless otherwise indicated, it is customary to assume that the domain is the largest subset of $\mathbb{R}$ where that formula is welldefined. Thus, if we write
$$f(x)=\frac{1}{x-1}$$
without other indications, then it is understood that $\mathcal{A}=\mathbb{R} \backslash{1}$ and $\mathcal{B}=\mathbb{R}$. In this case the image of $f$ is
$$\operatorname{Im}(f)=\mathbb{R} \backslash{0}$$
which is a set strictly contained in the codomain $\mathbb{R}$. It is also customary to say that $\mathbb{R} \backslash{1}$ is the natural domain of definition of the function $f$ above. Note that, it is possible to consider $f$ as a function defined on a subset of $\mathbb{R} \backslash{1}$. For example, one may consider the function $g$ defined from $\mathcal{A}={x \in \mathbb{R}: x>1}$ to $\mathcal{B}=\mathbb{R}$ by means of the same formula above, that is
$$g(x)=\frac{1}{x-1},$$
for all $x>1$. Note that
$$\operatorname{Im}(g)={y \in \mathbb{R}: \quad y>0} .$$

## 数学代写|微积分代写Calculus代考|Sets, sequences, functions

2.1 理论背景

2.1.1 集合的上限和下限

2.1.2 集合的最大值和最小值

2.1.3 集合的上确界和下确界

a) $x \leq L$ 对所有人 $x \in \mathcal{A}$ (IE， $L$ 是 个上限)；
b) 对所有人 $\epsilon>0$ 那里存在 $x \in \mathcal{A}$ 这样 $x>L-\epsilon\left(\mathrm{IE}^{\prime} ， L\right.$ 是最小上限 $) 。$

## 数学代写|微积分代写Calculus代考|Domain, codomain and image of a function

a) $x \geq l$ 对所有人 $x \in \mathcal{A}$ (IE， $l$ 是下界)；
b) 对所有人 $\epsilon>0$ 那里存在 $x \in \mathcal{A}$ 这样 $x1$ 至 $\mathcal{B}=\mathbb{R}$ 通过上面相同的公式，即
$$g(x)=\frac{1}{x-1},$$

$$\operatorname{Im}(g)=y \in \mathbb{R}: \quad y>0$$

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