1.5 上限，下限

定義 1.13 (上限，下限) 集合 $X\subset\mathbb{R}$ に対して

$\forall x\in X$ , $x\le M$ をみたす最小な $M\in\mathbb{R}$ が存在 $\Leftrightarrow$ $\sup X = M$
$\forall x\in X$ , $m \le x$ をみたす最大な $m\in\mathbb{R}$ が存在 $\Leftrightarrow$ $\inf X = m$

$\sup X$ を

の上限（supremum）， $\inf X$ を

の下限（infimum）という．

例 1.14 (上限，下限の具体例)

	$\displaystyle \sup\mathbb{N}=\infty,$		$\displaystyle \inf\mathbb{N}=1$
	$\displaystyle \sup (1,2] = 2,$		$\displaystyle \inf (1,2] = 1$
	$\displaystyle \sup \left\{\left.\,{(-1)^{n}}\,\,\right\vert\,\,{n\in\mathbb{N}}\,\right\}=1,$		$\displaystyle \inf \left\{\left.\,{(-1)^{n}}\,\,\right\vert\,\,{n\in\mathbb{N}}\,\right\}=-1$
	$\displaystyle \sup\left\{\left.\,{\frac{1}{2^n}}\,\,\right\vert\,\,{n=0,1,2,\cdots}\,\right\} = 1,$		$\displaystyle \inf\left\{\left.\,{\frac{1}{2^n}}\,\,\right\vert\,\,{n=0,1,2,\cdots}\,\right\} = 0$
	$\displaystyle \sup \left\{\left.\,{x^2}\,\,\right\vert\,\,{-1\le x\le 1}\,\right\}=1,$		$\displaystyle \inf \left\{\left.\,{x^2}\,\,\right\vert\,\,{-1\le x\le 1}\,\right\}=0$
	$\displaystyle \sup \left\{\left.\,{x^2}\,\,\right\vert\,\,{-1<x<1}\,\right\}=1,$		$\displaystyle \inf \left\{\left.\,{x^2}\,\,\right\vert\,\,{-1<x<1}\,\right\}=0$