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--- math104-s21:s:morganmakhina [2021/05/03 05:47]
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-==== **Real Analysis: Review & Sample Questions** ====
+=====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?**
@@ Line 6: / Line 115: @@
 **
 Topics Covered (with key definitions & theorems):**
+(This is a work in progress, and organization will improve soon!)
+) 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 $\mathbb Q$)
+) 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

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