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Gap lemma

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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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