10 - Approximation Property

Approximation Property

Let $S\subseteq \mathbb{R}$ be non-empty and bounded above
For any $ϵ>0$ there exists $s_e\in S$ such that

$$\sup S-ϵ<s_{ϵ}\le \sup S$$

TLDR: There exists $s_{ϵ}\in S$ that is arbitrarily close to $\sup S$

Proof

Note that by definition of the supremum we have

$$s\le \sup S\text{ for all }s\in S$$

Suppose for a contradiction that

$$\sup S-ϵ\ge s\text{ for all }s\in S$$

Then $\sup S-ϵ$ is an upper bound for $S$ but $\sup S-ϵ<\sup S$
But this is a contradiction so there

$$\text{exists }{s}_{ϵ}\in S\text{ with }\sup S-ϵ<s_{ϵ}$$