Real analysis things to learn

Types of proof - direct, contradiction, induction

The principle of induction

What is a set? What are the symbols to show an element of a set?

Union & intersection definitions

Empty set definition

Subset definition & when are two sets equal

Function definition - domain and range

The modulus/absolute value definition

Rules of modulus (multiplicativity, triangle inequality)

The Archimedian axiom

Theorem (Density of Q in R)

Upper bound definition

Supremum definition

Axiom of Completeness

sup(c+A) = ??

s=supA if and only if ?

Theorem (Nested Interval Property)

Theorem (Existence of square roots)

Theorem 1.11 (Existence of root 2)

Definition 2.1 (A sequence of real numbers is…)

Definition 2.3 (Convergence of a sequence)

Definition 2.4 (Convergence of a sequence: geometric version)

Template for a formal proof that xn→n (convergence)

Proposition 2.8 (Uniqueness of limits)

Definition 2.10 (A sequence that does not converge is said to…)

Definition 2.11 (When is a sequence of real numbers called bounded?)

Proposition 2.12 (Every convergent sequence is…)

Theorem 2.13 (The Algebraic Limit Theorem)

Theorem 2.14 (The Order Limit Theorem)

Definition 2.15 (When is a sequence increasing/decreasing/neither?)

Theorem 2.16 (Monotone Convergence Theorem)

Bernoulli’s inequality

Subsequence definition

Theorem 2.19 (Subsequences of a convergent sequence converge to…)

Theorem 2.22 (Bolzano-Weierstarß)

Definition 2.23 (Cauchy sequence)

Theorem 2.24 (Cauchy Criterion)

Lemma 2.25 (Cauchy sequences - bounded?)

Theorem 2.24 (Cauchy Criterion)

Definition 3.2 (Infinite series)

The Algebraic Limit Theorem for Series

Theorem (Cauchy Criterion for Series)

Corollary (If ? converges then ?)

Theorem (Comparison Test)

Definition (When is an infinite series absolutely convergent?)

Theorem (Absolute Convergence Test)

Theorem (Alternating Series Test)

Definition - when is a function is continuous?

Theorem - Continuity in terms of sequences

Corollary (Criterion for discontinuity)

Theorem (Algebraic Continuity Theorem)

Theorem (Composition of Continuous Functions)

Theorem (The Intermediate Value Theorem)

Definition (Limit point)

Definition limit point formal writing

Theorem (Sequential Criterion for Functional Limits)

Theorem (Algebraic Limit Theorem for Functional Limits)

Definition - differentiable functions and the derivative

Alternate way for writing the derivative

Theorem - when a function is differentiable equivalence

Corollary (Differentiable functions are continuous)

Theorem (The Algebraic Differentiability Theorem)

Theorem (Chain Rule)

Definition (Local Minimum and local maximum)

Theorem (Fermat’s Theorem/Interior Extremum Theorem)

Corollary (Rolle’s Theorem)

Theorem (The Mean Value Theorem)

Corollary - with interval and differentiability

Definition 5.15 (Monotonically increasing/decreasing, strictly increasing/decreasing)

Corollary 5.16 (monotonically increasing/decreasing, strictly increasing/decreasing - differentiable functions)

Theorem 5.17 (The Generalised Mean Value Theorem)

Theorem 5.18 (l’Hospital)