Real Analysis Exam 1 Review Problems and Solutions

#realanalysis #realanalysisreview #realanalysisexam

Links and resources
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⏱️TIMESTAMPS⏱️
(0:00) Introduction
(0:37) Define supremum of a nonempty set of real numbers that is bounded above
(3:43) Completeness Axiom of the real numbers R
(5:43) Define convergence of a sequence of real numbers to a real number L
(9:06) Negation of convergence definition
(11:42) Cauchy sequence definition
(13:37) Cauchy convergence criterion
(15:39) Bolzano-Weierstrass Theorem
(18:58) Density of Q in R (and R - Q in R)
(19:37) Cardinality (countable vs uncountable sets)
(21:26) Archimedean property
(23:40) Subsequences, limsup, and liminf
(28:43) Prove sup(a,b) = b
(35:34) Prove a finite set of real numbers contains its supremum
(39:39) Find the limit of a bounded monotone increasing recursively defined sequence
(45:28) Prove the limit of the sum of two convergent sequences is the sum of their limits
(51:28) Use completeness to prove a monotone decreasing sequence that is bounded below converges
(58:14) Prove {8n/(4n+3)} is a Cauchy sequence

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[algorithmo]

Please make more of this. It is amazing! Can you also cover the full book of rudin?

[TepsiMorphic]

Finding this exam review playlist 9 days before my exam was a miracle. Than you so much!

[neelg]

This is incredible, I honestly have not found something so reflective of my exam tomorrow on Baby Rudin Ch 1-3.

[SamSarwat]

Problem 18: Its a convergent sequence (limit being 2). Since convergence and Cauchy are equivalent in the real line, the sequence is Cauchy.

[brandonfox]

For the Bolzano-Weierstrass Theorem, I would've emphasized more on the fact that the existing subsequence not only converges, but that it's monotone, too.

[stevemenegaz]

This was very pleasant to watch. You have a very nice teaching style. Thank you. I plan to watch more.

[jademath]

Great video. I learned a lot. Thank you, sir. Watching from Ph.

[ronnies.]

Wish I had this video when I took Real Analysis! Would’ve been super helpful. That class was rough.

[nigellbutlerrr]

I prefer the real analysis of Advanced Engineering maths, by Kreyzig and, R V Churchill texts.

[skapun]

I am comfortable working with definitions and seeing examples to illustrate the meaning of the definition. However, when I get to proving things (I have taken a proofs class in the past), outside of stating obvious definitions and relevant theorems, I struggle to complete the rest of the proof and if I do, it is most likely incorrect. What do you suggest I do to improve on this besides doing more questions?

[forheuristiclifeksh]

1:00 Please also make every math subject!!!! THX!!

[UliRaudales]

I am currently taking Real Analysis with Russell Gordon 😂

[mathematicaleconomist]

What sounds hardest to your ears? 1. Masters Degree in Economics 2. Masters Degree in Applied Mathematics 3. Masters Degree in Statistics? Which of the 3 sounds easiest to your ears?

[thehardlife]

I did real analysis back in 2011 and i got out my old lecture notes, the proofs i find hard particularly the one of prove sup(A +b) = sup(A) +sup(B), i know you have done a video on that one, but the proof just seems circular in arguments

[MichaelMarteens]

Thanks! It was very helpful to study with you.

[thehardlife]

At 27:00 i was wondering on Wikipedia it says that the limsup as n goes to infinity is the inf of the set of supremums, so i am just confused why you chose 8 as the inf of the supremum would be - 2?

[trongtue]

May I ask I hear everyone said that in the higher Math there will be more proof and less dealing with number or Computational. And in that level the Math will be similar to philosophy because it will be more abstraction and idealist. I don't know that opinion right or not so can you explain for me a little bit more

[thehardlife]

The second part of the definition of supremum doesnt hold for sets such as (0, 1) U {2}

[samueldarko]

I have a paper in two hours. Wish me luck

[abdulrehmanbilal]

Can you please share the pdf file of this exam?