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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Interchanging limits
Exchanging limits, derivatives or integrals requires hypotheses such as suitable uniform convergence.
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Pointwise convergence
A sequence of functions converges separately at each point, with tolerances allowed to depend on the point.
Power series
A power series converges inside its convergence radius; boundary points require separate tests.
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.
Compactness
A space is compact when every open cover admits a finite subcover.
Knowledge connections
- Power series · Related · Map context
- Pointwise convergence · Related · Map context