Analysis I

Year 1 · 7 sections · 104 concepts · 373 practice questions

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What's covered

free1 · The real numbers7 concepts

Definition 1.1 · ℝ is a complete ordered field
Definition 1.2 · Absolute value and ε-neighbourhoods
Proposition 1.3 · Suprema and the ε-characterisation
Theorem 1.4 · The Archimedean property
Theorem 1.5 · Density of ℚ and of the irrationals
Theorem 1.6 · The nested interval property
Check 1.7 · Check: proving a sup or inf

2 · Sequences18 concepts

Definition 2.1 · Convergence of a sequence
Theorem 2.2 · Limits are unique
Proposition 2.3 · Convergent sequences are bounded
Theorem 2.4 · The algebra of limits
Proposition 2.5 · Limits preserve weak inequalities
Theorem 2.6 · The sandwich rule
Proposition 2.7 · Standard limits
Proposition 2.8 · The ratio criterion for sequences
Theorem 2.9 · The monotone convergence theorem
Example 2.10 · A recursively defined sequence
Example 2.11 · The number e
Proposition 2.12 · Subsequences
Lemma 2.13 · Every sequence has a monotone subsequence
Theorem 2.14 · The Bolzano–Weierstrass theorem
Theorem 2.15 · Cauchy sequences converge
Definition 2.16 · Divergence to infinity
Definition 2.17 · Limit superior and limit inferior
Check 2.18 · Check: writing an ε–N proof

3 · Series19 concepts

Definition 3.1 · Series and their sums
Proposition 3.2 · The nth-term test
Proposition 3.3 · Geometric series
Theorem 3.4 · The harmonic series diverges
Proposition 3.5 · Linearity and tails
Theorem 3.6 · Series of non-negative terms
Theorem 3.7 · The comparison test
Theorem 3.8 · The limit comparison test
Theorem 3.9 · Cauchy condensation
Corollary 3.10 · The p-series
Theorem 3.11 · The ratio test
Theorem 3.12 · The root test
Theorem 3.13 · The integral test
Theorem 3.14 · The alternating series test
Theorem 3.15 · Absolute convergence implies convergence
Theorem 3.16 · Rearrangements
Proposition 3.17 · The Cauchy criterion for series
Proposition 3.18 · e as a series, and e is irrational
Check 3.19 · Check: which convergence test?

4 · Limits and continuity16 concepts

Definition 4.1 · Limits of functions
Theorem 4.2 · The sequential characterisation of limits
Proposition 4.3 · Algebra of limits and the sandwich rule for functions
Definition 4.4 · One-sided limits
Definition 4.5 · Continuity
Proposition 4.6 · Combining continuous functions
Example 4.7 · Continuous and discontinuous functions
Theorem 4.8 · The intermediate value theorem
Corollary 4.9 · Fixed points and nth roots
Theorem 4.10 · Continuous functions on [a, b] are bounded
Theorem 4.11 · The extreme value theorem
Theorem 4.12 · Continuous injective functions and their inverses
Theorem 4.13 · Uniform continuity on closed bounded intervals
Definition 4.14 · Limits at infinity and infinite limits
Proposition 4.15 · sin x / x → 1
Check 4.16 · Check: proving continuity or discontinuity

5 · Differentiation16 concepts

Definition 5.1 · The derivative
Proposition 5.2 · Differentiable implies continuous
Theorem 5.3 · Sum, product and quotient rules
Theorem 5.4 · The chain rule
Theorem 5.5 · Derivative of an inverse function
Example 5.6 · Strange derivatives
Theorem 5.7 · Interior extrema have zero derivative
Theorem 5.8 · Rolle's theorem
Theorem 5.9 · The mean value theorem
Corollary 5.10 · Consequences of the mean value theorem
Example 5.11 · Inequalities from the mean value theorem
Theorem 5.12 · Cauchy's mean value theorem
Theorem 5.13 · L'Hôpital's rule
Theorem 5.14 · Taylor's theorem with Lagrange remainder
Corollary 5.15 · The second derivative test
Check 5.16 · Check: differentiability and using the MVT

6 · Integration17 concepts

Definition 6.1 · Partitions and Darboux sums
Lemma 6.2 · Refinements and the order of sums
Definition 6.3 · The Riemann integral
Theorem 6.4 · The Riemann criterion
Example 6.5 · A non-integrable function
Theorem 6.6 · Monotone functions are integrable
Theorem 6.7 · Continuous functions are integrable
Proposition 6.8 · Properties of the integral
Proposition 6.9 · Finitely many discontinuities don't matter
Theorem 6.10 · Mean value theorem for integrals
Theorem 6.11 · The fundamental theorem of calculus (part 1)
Theorem 6.12 · The fundamental theorem of calculus (part 2)
Theorem 6.13 · Integration by parts
Theorem 6.14 · Integration by substitution
Proposition 6.15 · Riemann sums converge to the integral
Definition 6.16 · Improper integrals
Check 6.17 · Check: integrability and evaluating integrals

7 · Power series11 concepts

Theorem 7.1 · The radius of convergence
Proposition 7.2 · Computing the radius of convergence
Example 7.3 · Behaviour at the endpoints
Theorem 7.4 · Power series can be differentiated term by term
Definition 7.5 · The exponential function
Proposition 7.6 · Properties of the exponential
Definition 7.7 · Sine and cosine as power series
Proposition 7.8 · Properties of sine and cosine
Theorem 7.9 · Taylor series
Example 7.10 · Series for log and arctan
Check 7.11 · Check: working with power series

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Proposition 1.3 · Suprema and the ε-characterisation

$u = \sup S$ : upper bound, and every $u - \varepsilon$ is beaten by some element. There's always a sequence in $S$ converging to $\sup S$ . Comparisons: bigger set ⇒ bigger sup; "every $s\le$ every $t$ " ⇒ $\sup S\le\inf T$ .

Proof

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