Gap lemma
Revision as of 03:44, 15 February 2008 by Shreyas patankar (talk | contribs) (New page: '''Gap lemma''' is actually a trivial corollary of the completeness property of <math>\mathbb{R}</math> but is extremely useful in real analysis ==Statement== Let <m...)
Gap lemma is actually a trivial corollary of the completeness property of but is extremely useful in real analysis
Statement
Let be bounded above
Let
Then, such that
Proof
Assume if possible,
such that
Consider
We see that is an upper bound of
, but
which contradicts the assumption that
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