=====Morgan's Real Analysis Review Page=====

====Number systems:====

**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)?
**



====Sets, sequences, series...====

**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?**

====Topology:====

**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?**

====Continuity:====

**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?**

====Sequences of Functions:====

**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?
**

==== Derivatives:====

**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?**


====Integration:====




====Extras:====


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))**


====Bonus Questions:====

**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