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math104-s21:s:morganmakhina [2021/05/05 05:58]
107.77.212.37 		math104-s21:s:morganmakhina [2026/02/21 14:41] (current)
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-====Morgan's Real Analysis Review Page====+=====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?**
  
  
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 (This is a work in progress, and organization will improve soon!) (This is a work in progress, and organization will improve soon!)
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 1) Number systems: $\N$, $\Z$, $\mathbb Q$, $\R$, $\C$, others, & some of their properties 1) Number systems: $\N$, $\Z$, $\mathbb Q$, $\R$, $\C$, others, & some of their properties
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 Every convergent sequence has a monotone subsequence Every convergent sequence has a monotone subsequence
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math104-s21/s/morganmakhina.1620194297.txt.gz · Last modified: 2026/02/21 14:44 (external edit)

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