About Me: I'm a first year math major from Nevada. I enjoy playing guitar and running.

Course Notes

• Overview of Course
• Discussed natural numbers, integers, and rationals
• Problem with rationals: has holes which prevent us from getting sharp bounds on subsets
• Exercises

• Rational Zeros Theorem
• Construction of $\R$ from $\Q$: Dedekind Cuts vs. Cauchy Sequences
• Completeness Axiom + Archimedean Property for $\R$
• Definition of limits/convergence for sequences

• Showed Convergence Sequences are bounded.
• Defined operations on convergent sequences.
• Showed some useful limits.

• Monotone Sequences
• Recursive Definition of Sequences
• lim inf and lim sup of a Sequence

• Cauchy sequences
• Cauchy sequences always converge in $\R$
• Subseqeunces
• Cantor's Diagonal trick to produce a convergent subsequence

• All sequences have a monotone subsequence
• All bounded sequences have a convergent (monotone) subsequence
• If $S$ is the set of subsequential limits of $s_n$, then sup$S$ = limsup$s_n$ and inf$S$ = liminf$s_n$

• limsup(a_nb_n) = lim(a_n)limsup(b_n) for convergent series $a_n$ with limit greater than 0
• Introduced Series
• “Sanity Check” and Comparison Test
• Root and Ratio Tests

• Series
• Summation by Parts
• Power Series

• What is a good way to approach coming up with inequalities to use in proof, as in the Rudin exercises this week.
• What are some good counterintuitive counterexamples to keep in mind when working on problems.
• What specific properties of absolute convergence should we be familiar with for the exam, eg. rearrangements etc.
• What properties does multiplication in limsup(a_nb_n) have in general.
• Is there a good way to get intuition for accumulation of infinite series, eg. the case of sum(1\n)

• Definition of Metric Space + examples
• Topology
• Open Sets

• More Metric Space examples
• Sequences + Cauchy Criterion
• Closure/ Closed Sets

• Continuous Maps (open cover def and sequential def)
• Inherited Topology

• Open cover compactness
• Sequential compactness

• Sequential Compactness $\to$ Open Cover Compactness

• Connectedness

• Continuous maps preserve compactness and connectedness
• Uniform Continuity
• Discontinuity

• Sequences and Series of Functions
• Uniform Convergence

• Differentiation
• Rolle's Theorem

• Generalized Mean Value Theorem
• L'Hopital's rule

• Higher Derivatives
• Taylor's Theorem

• Taylor Series
• Power Series
• Reimann Integral

• Integration
• Reimann - Stieltjes Integral

• Reimann - Stieltjes Integral

• Properties of Integrals

• Uniform Convergence with Integration
• Uniform Convergence with Differentiation

• Hw 1
• Hw 2
• Hw 3
• Hw 4
• Hw 5
• Hw 6
• Hw 7
• Hw 8
• Hw 9
• Hw 10
• Hw 11