Lecture Notes

1-5) What are real numbers, anyway? Why do we need them? How can we rigorously define (ie, construct) them? What are some properties of $\R$ that other number systems don't have? And, by the way: what are some properties of $\N$ that we use in real analysis (perhaps sometimes taking them for granted)?

6) What's the difference between sets, sequences, and series??

7)What's a Cauchy Sequence?

8) How can you tell if a sequence converges (and, if it does, how can you tell what it converges to)?

9) How can you tell if a series converges (and, if it does, how can you tell what it converges to)?

10)What is radius of convergence?

11) Why do we care about monotone sequences?

What's a metric space?

What are some familiar and less familiar metrics (distance functions)?

What are some examples of functions that aren't distance functions, even though they have some properties in common with distance functions?

What's a complete metric space?

What are topological spaces, and how is this notion different from that of a metric space?

Topological concepts are intuitive… until they're not. What are some caveats to watch out for?

Which properties of topological subspaces depend on the ambient space, and which do not?

Compact vs closed & bounded: when are these equivalent? When are they not equivalent?

What does “sequentially compact” mean, and when is this property equivalent to compactness?

What's so special about compact sets? (Ie, what are some theorems we proved about compact sets that won't hold for other kinds of sets?)

What is the Heine-Borel Theorem? When can we apply it, and when should we not apply it?

What is the Bolzano-Weierstrass Theorem, and how does it relate to the Heine-Borel Theorem?

What are some of the particularly useful results in this section?

What's a continuous function?

What conclusions can we make if we know a function is continuous?

What conclusions might we be tempted to make about continuous functions that actually aren't true?

What is uniform continuity?

What is the difference between pointwise and uniform convergence?

What are some examples of sequences of functions that converge pointwise but not uniformly?

What conclusions can we make about uniformly converging sequences of functions that would no longer necessarily be valid if we replaced uniform convergence by pointwise convergence?

What conclusions might we be tempted to make about uniformly converging sequences (of functions) that aren't actually true?

What are some of the key theorems in this section?

What are some surprising results in this section?

When do Taylor series approximations fail?

What is Taylor's Theorem, and why is it useful?

Note: The following questions appeared on Anton's review page (which I found very useful!). I found them too important to omit, and too well-stated to paraphrase. The two exam-related questions were ones that I did not get right on the midterm.

What is the Weierstrass M-Test?

Why is the set $[0,1] \cap \mathbb{Q}$ not compact while $[0,1]$ is? (MT2, Q1, (4))

Why is the set $\{0\} \cup \{1/n | n \in \mathbb{N}\}$ compact? (MT2, Q1, (5))

What were some of the particularly surprising, memorable, and fun things I learned in this course?

Briefly list the most significant concepts/theorems covered in this course.

Where can I find some sample exams to do for practice?

Topics Covered (with key definitions & theorems):

(This is a work in progress, and organization will improve soon!)

1) Number systems: $\N$, $\Z$, $\mathbb Q$, $\R$, $\C$, others, & some of their properties

Archimedian Property

(Something we regrettably skipped: Dedekind's construction of $\R$ from $\mathbb Q$)

2) Max, min, upper bound, lower bound, sup, inf defined.

Completeness Axiom of $\R$: Every nonempty subset of $\R$ that's bounded from above has a least upper bound in $\R$ (+ analogous result for greatest lower bound)

Sequences and their limits

(epsilon & N definition of limit)

Some nice theorems about properties of limits, which we can use in lieu of the epsilon & N definition to quickly establish convergence (or non-convergence) . . . Cauchy sequences defined

Monotone sequences

Theorem: All bounded monotone sequences are convergent.

Theorem: As it turns out, Cauchy sequences are precisely the sequences that converge - i.e., we can use the Cauchy criterion as an equivalent definition of convergence. (Sometimes one definition is easier to work with than another in writing a proof, so this is good news).

lim inf, lim sup of a sequence (Thm: all bounded sequences have them)

Recursive sequences, & tricks for finding their limits, if extant (see Feb 4 note) (cobweb diagram)

Subsequences:

Every convergent sequence has a monotone subsequence