Proposition 6.2 Bolzano-Weierstrass 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