Looking for learning material for this real analysis practical exam.

[marble plinth]

Hi, this is one of the examples of how an exam looks. What I am looking for are practice books/videos/playlists which can cover these advanced examples. My literature covering this course is very bad with limiting and easy tasks.

So far, I looked into a few analysis books including:

Abbott - Understanding Analysis
Spivak - Calculus
Kenneth Ross - Real Analysis

And a bunch of books related to my native language but I couldnt find any difficult tasks like these. Even when there are some, theres no detailed explanation covering it.
I know everything about basics of Analysis, but what I struggle with are advanced tasks that also require employing theory or require a completely different mindset from usual tasks.
Any tips you can share about learning these challenging tasks is more than welcome! Please share any yt playlist/book that has good explanation to majority of these, thanks! (I watched a lot of stuff on youtube, but its all usually basic introductory stuff I know already...)

[near moss]

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[blazing thicket]

have you tried searching up random analysis I exam papers?

i googled "real analysis I exam" and clicked the first link

try these

[blazing thicket]

these are all very intuitive results, where are you exactly having trouble?

for theory, folland's book is preety good too

[marble plinth]

yeah, but everything I found was either easier or harder like yours (full proof exam). Mine has this sort of mixture in between and what I found challenging are the examples that are set up.

[blazing thicket]

how did you find them challenging?

in what way?

here are some potential issues :

• Being unable to start a problem, feeling lost.
• Lack of knowledge with usual theory
• Trouble doing them in the exam's time limit

[marble plinth]

we havent gone through many various examples. So 2nd and 3rd tasks are both completely new to me in the sense of their formulation. We only did basic recurrent sequences

[blazing thicket]

how much time do you have till the exam

what's your syllabus

[marble plinth]

Id say its mostly the first one, we havent done many various examples and thats why im seeing something completely new

[marble plinth]

3hrs

[blazing thicket]

personally, abbot and Ross aren't my preferred texts

[marble plinth]

this one is called real analysis

[blazing thicket]

till the exam not during

when is the exam?

[marble plinth]

during exam

[marble plinth]

15 days from now

[blazing thicket]

oh

you have enough time

here's what you can do

[marble plinth]

true but as I said whatever I find online is usually covering basic stuff I already know so I feel like im wasting time

[blazing thicket]

you already know the basics of real analysis correct?

[marble plinth]

yes, I only struggle with some differentiation and series theory

[blazing thicket]

ill name some things tell me if you've done it or not it'll help me assess where you are at

[marble plinth]

sure

[blazing thicket]

countablity/uncountablity

have you done it?

[marble plinth]

not sure what you mean by it

do you mean in sets?

[blazing thicket]

yes

[marble plinth]

in sets we only did supremum, infimum, min, max etc

so I guess in a sense yes

if its uncountable it doesnt have a maximum but has a supremum for example

is my analogy correct?

[blazing thicket]

this is incorrect

[0,1] has a maximum, it's 1

[marble plinth]

yeah I see

[blazing thicket]

have you done metric spaces?

or the more topology-ish aspects

[marble plinth]

only slightly in theory

[blazing thicket]

like compact, closed, open, complete

etc

[marble plinth]

yeah

[blazing thicket]

is your course just real analysis or real analysis I?

[marble plinth]

real analysis 1

[blazing thicket]

ok

[marble plinth]

second course has more deeper introduction to metric spaces

[blazing thicket]

i assume you've done continuity and the differentiation

[marble plinth]

yes

[blazing thicket]

judging from this exam

[marble plinth]

we also did taylor expansion, maclaurin

[blazing thicket]

how do you feel about the paper i sent?

[marble plinth]

seems a bit harder but involves only theory

[blazing thicket]

can you do problem 2 and 6

how would you start them

for reference

[marble plinth]

2 looks familiar but I have no intuition for it. 6 is what Ive seen multiple times but not really sure how to do

[blazing thicket]

now try to do the problems

try to apply things you did in class

if you are hardstuck (spent 1-2 hours with little to no progress) look at the solution

you'll learn something new and develop intuition too

[blazing thicket]

this goes for nearly anything you are trying to better at mostly

goldmine of material

https://www.math.ucla.edu/~steven/quals/analysis/

[marble plinth]

Ill check it out thx

[weak plume]

@marble plinth has given 1 rep to @blazing thicket

[blazing thicket]

@marble plinth lemme know how far you've come with the problems btw

[marble plinth]

alright will do

okay so for 2. I see that the $x_0$ is either going to be 0 or 1 and from there sequence is monotonically increasing or decreasing but I havent done any proofs regarding it. Regarding proof part, I guess upon contstructing a sequence I could take 0 and 1 as limits and perform epsilon proof for both, but how to get to formula for the sequence is hard part for me

[pine plaza]

danilojonic

[marble plinth]

What I thought was to reverse engineer my knowledge in this by setting some parameter $L = \displaystyle \lim_{n \to \infty}x_n$

[pine plaza]

danilojonic

[marble plinth]

I already know the limits are 0 and 1, so $L=0 \wedge L=1$

[pine plaza]

danilojonic

[marble plinth]

so this would be $x_n(x_n - 1)$

[pine plaza]

danilojonic

[blazing thicket]

wdym

like a recurring sequence x(n+1)=x(n)(x(n)-1)?

not sure this hits all numbers in the interval [0,1]

heres a hint (you can ask me for hints generally if I'm online, they are great)

If the limit points of the sequence are dense in [0,1] what can we say about this set?

[marble plinth]

maybe $x_0 = \frac{1}{2}$? But again this is just a speculation. If progressively more and more sequence elements move towards 0 and 1, Id assume it would be in a similar way to how $x_n = (-1)^n$ works but that sequence is divergent so maybe this one is too?

[pine plaza]

danilojonic

[blazing thicket]

we need it's limit point set to be all reals in [0,1] danil

x_n=(-1)^n just goes -1 and 1 -1 and 1

[marble plinth]

thats what I was about to say yeah

[blazing thicket]

well i just wanted to see how you try to approach it

from what i see, you just need more practice solving problems as you mentioned

[marble plinth]

so the sequence doesnt diverge but it includes 0 and 1

[blazing thicket]

make a new help post for this if you need help

and keep doing the papers