Gap lemma

Gap lemma is actually a trivial corollary of the completeness property of $\mathbb{R}$ but is extremely useful in real analysis

Statement

Let $A\subset\mathbb{R}$ be bounded above

Let $u=\sup{A}$

Then, $\forall\epsilon>0\;\;\exists a\in A$ such that $|u-a|<\epsilon$

Proof

Assume if possible, $\exists$ $\delta>0$ such that $|u-a|>\delta$ $\forall$ $a\in A$

Consider $u'=u-\delta$

We see that $u'$ is an upper bound of $A$ , but $u'<u$ which contradicts the assumption that $u=\sup{A}$ This article is a stub. Help us out by expanding it.

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Gap lemma

Statement

Proof