04

題目

Show that $d = \inf(S)$ iff $d$ is a lower bound for $S$ and for any $\epsilon > 0$ there is an $x\in S$ such that $d \geq x-\epsilon$

解答

$\implies$

It's trivial that $d$ is a lower bound. Choose $x = d + \frac{\epsilon}{2}$, then $d\geq d-\frac{\epsilon}{2}=x-\epsilon.$

$\impliedby$

Assume $d^{\prime}$ is another lower bound and $d^{\prime} > d$. Choose $\epsilon = \frac{d^{\prime}-d}{2}$, there is an $x\in S$ such that

$$x \leq d+\epsilon = \frac{d^{\prime}+d}{2} < d^{\prime},$$

which leads to a contradiction. Then $d$ is the $\inf(S)$.