Analysis WebNotes
Proposition 6.1 Subsequences of converging sequences

Proposition 6.2 Bolzano-Weierstrass theorem

Ramsey's Theorem

Corollary 6.3 Closed bounded sets of real numbers are sequentially compact

Proposition 6.4 Sequentially compact sets are closed and bounded

Theorem 6.5 Sequentially compact sets in Euclidean spaces

Theorem 6.6 Continuous images of sequentially compact sets are sequentially compact

Corollary 6.7 Continuous images of closed bounded sets in Euclidean spaces

Corollary 6.8 Continuous functions on sequ compact sets are bounded

Lemma 6.9 Closed subsets of sequentially compact are sequentially compact

Proposition 6.10Inverses of continuous bijections from sequentially compact sets

Proposition 6.11 Continuous functions have maximimums on sequentially compact sets

Proposition 6.11a Continuous and compact functions

Corollary 6.12 Continuous functions also attain their minimums

Lemma 6.13 A differentiable function has zero derivative at a maximum

Proposition 6.14 Rolle's Theorem

Theorem 6.15 Mean Value Theorem

Corollary 6.16 A differentiable function is increasing iff its derivative is non-negative

Corollary 6.17 A function with a strictly positive derivative is strictly increasing

Proposition 6.18 Inverse Function Theorem

Corollary 6.19 Power Rule for Derivatives

Proposition 6.20 Differentiable Bounded Functions are Uniformly Continuous

Proposition 6.21 Uniformly Continuous Metric Spaces

Proposition 6.21a Uniformly Continuous Compact Sets

Lemma 6.22 Intersection of Closed Subsets of Compact Sets

Lemma 6.23 Compact sets are closed

Theorem 6.24 Metric Spaces are Compact iff they are Sequentially Compact

Lemma 6.25 Finite Sets of Points in Sequentially Compact Sets

Theorem 6.26 Sequences in Compact Metric Spaces Converge iff they are Cauchy

Corollary 6.27 A sequence of real or complex numbers converges if and only if it is a Cauchy Sequence