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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Metric spaces
A metric assigns nonnegative distances satisfying separation, symmetry and the triangle inequality.
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Compactness
A space is compact when every open cover admits a finite subcover.
Extreme value theorem
A continuous real function on a nonempty compact set attains its maximum and minimum.
Bolzano–Weierstrass theorem
Every bounded sequence in finite-dimensional Euclidean space has a convergent subsequence.
Cauchy sequences
Terms of a Cauchy sequence eventually become arbitrarily close to each other.
Completeness of the real numbers
Every nonempty real set bounded above has a least upper bound.
Knowledge connections
- Extreme value theorem · Related · Map context
- Compactness · Related · Map context