Mathematical and Real Analysis
Explore real numbers and limits; continuity and compactness; functions and integration. Concepts and connections are available independently of video coverage.
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Cauchy sequences
Terms of a Cauchy sequence eventually become arbitrarily close to each other.
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Limit superior and inferior
Tail suprema and infima describe the extreme limiting behavior of a sequence.
Sequence convergence
A sequence converges when its terms eventually lie within every prescribed distance of a limit.
Bolzano–Weierstrass theorem
Every bounded sequence in finite-dimensional Euclidean space has a convergent subsequence.
Compactness
A space is compact when every open cover admits a finite subcover.
Completeness of the real numbers
Every nonempty real set bounded above has a least upper bound.
Knowledge connections
- Sequence convergence · Related · Map context
- Limit superior and inferior · Related · Map context
- Sequence convergence · Prerequisite · Map context