# 0 - Some notes on real analysis.
[[notes/1-dedekind-completeness-archimedean-denseness-of-q|1 - Dedekind completeness, Archimedean, and denseness of $\mathbf{Q}$.]]
- The reals are the essentially unique Dedekind complete ordered field.
- Any ordered field contains a copy of the rationals.
- The rationals are not Dedekind complete.
- The rationals are Archimedean.
- An order field that is Archimedean if and only if the rationals are dense in it.
- The reals are Archimedean, hence rationals are dense in the reals.
- There exists non-Archimedean ordered fields.
- The reals are the maximally Archimedean ordered field, and any Archimedean ordered field embeds into the reals.
- Any two Dedekind complete ordered fields are isomorphic, and a Dedekind complete ordered field can be constructed, say by Dedekind cuts.
[[notes/2-more-consequences-of-the-reals|2 - More consequences of the reals.]]
- Absolute value and triangle inequality.
- Sequences and convergence in an ordered field.
- Monotone completeness.
- Nested completeness.
- Bolzano and Weierstrass completeness
- Cauchy completeness
- Cut completeness
[[notes/3-analysis-in-reverse|3 - Analysis in reverse.]]
- Are the notions of completeness equivalent?
- Calculus theorems as axioms.
[[notes/4-encyclopedic-sequences|4 - Encyclopedic sequences. ]]
- How fast can a sequence grow to be encyclopedic?
- Density of irrational multiples. (Dirichlet)