# 解析学の基本的な定義

## 上界、下界の定義

$A$ is a set.
Upper bound of $A$ (= $U(A)$) and lower bound of $A$ (= $L(A)$) is defined by followings.
$$U(A) = \{b|b\in \mathbb{R},\forall a\in A,a\le b\}$$
$$L(A) = \{c|c\in \mathbb{R},\forall a\in A,c\le a\}$$

## sup,infの定義

$\sup A$ and $\inf A$ is defined by followings.
$$\sup A \in U(A), \forall a \in U(A), \sup A \le a$$
$$\inf A \in L(A), \forall a \in L(A), a \le \inf A$$

## 有界の定義

$A$が上に有界($A$ is bounded above)とは
$$U(A) \neq \emptyset$$
$A$が下に有界($A$ is bounded below)とは
$$L(A) \neq \emptyset$$
$A$が有界($A$ is bounded)とは
$$L(A) \neq \emptyset \text{ and } U(A) \neq \emptyset$$