Problems in Real Analysis | Ep. 1

[mideoryan]

“…problems range from eadium, medium…EASY medium”

That cracked me up for some reason, so adorable

[amtsaal]

it's really fun to see math guy doing his work

[stefanpostnikov]

Very inspiring and cosy videos. Really like it.

[foobar]

I appreciated real analysis much more when I learned rigorous probability theory. It's similar ideas but somehow the change in perspective makes certain things more intuitive at least for me

[shuhaotian]

The second problem seems a lot less intimidating if you substitute g = f - 1 into the given integrals from the start (which is natural since the condition f ≥ 1 becomes g ≥ 0 which is easier to work with). The integrals in the qn statement becomes ∫ g dx = 1, ∫ g^2 dx = 2 and ∫ g^3 dx ≥ 4. This makes it a lot easier to guess how to use cauchy-schwarz, since the numbers 1, 2, 4 are so much nicer than the given 2, 5, 14
You can also use cauchy-schwarz in the form ( ∫ (1 · g) g dx )^2 ≤ ( ∫ (1 · 1) g dx )( ∫ (g · g) g dx ) rather than ( ∫ g^1/2 · g^3/2 dx )^2 ≤ ( ∫ g^1/2 · g^1/2 dx )( ∫ g^3/2 · g^3/2 dx ), i.e. using the inner product <a, b>=∫ (a · b) g dx rather than <a, b>=∫ a · b dx
Everything here is equivalent to the stuff in the vid, but it makes spotting the solution much more natural than trying to notice (f-1)^2=(f-1)^1/2(f-1)^3/2 and using cauchy directly

[benjaminvuchetich]

My favorite math channel cuz he keeps it real

[wowZhenek]

Man ... I feel depressed about problem 2 ... this just feels so ... ugh ... so bullshit of a problem if you don't know the trick ... this just feels so lame, so contrived ...I tried solving it by adding and subtracting summands to complete the (f - 1)^3 and then I ended up with

integral (f^3) dx = integral (f - 1)^3 dx + 10.

I wish this led anywhere, but sadly it does not.

[aryansaxena]

addictive channel my man, appreciate the effort you put in <3

[amadeusamadeus]

I don’t if this an odd request or not but would consider doing a video on your studying routine? Like how do you organize your daily life and manage between studying and reading?

[hubomba]

Things like the second problem is why I always preferred Algebra. Analysis seems to rely on conjuring up a bunch of specific constructions to continue with a lot of the proofs that seem impossible to see unless you spend a lot of time on it or know it a priori.

[tusharverma]

Hi great video!!
Can you please share the name off the pen you used??

[swaruppaul]

This is a great video! Planning to cover this series within a few days. It would of tremendous help if you (re)start a series like this again, where you do few selected problems from such core areas of math.
These are actually fantastic learning resource for students like me who are learning math in isolation and doesn't have any resource person to talk to... You're like a friend from whom we can learn new stuff through discussion. Thank you for the video, today I learned about Convergence in measure, Chebyshev inq and Markov's inequality and a great problem from you!

[爸爸到底-sx]

I think the key for the not so obvious direction of the third problem is to observe that min(1, |f-f_n|) is a dominated sequence of measurable functions. So convergence in measure implies convergence in the L^1 norm.

[lugia]

Please do a number theory video would be interesting to see your thought process in solving those sorts of abstract problems

[ares]

Hi, first of all I love your videos! I just finished my 1st semester of CS in Germany, but I've enjoyed the math modules (analysis and discrete structures) a lot more than the actual CS modules, and I am considering switching my major after the 2nd to math, as there won't be many more math classes to come and I won't be able to choose them later on either, but I am unsure if it is "worth" it, financially speaking - job prospects as a mathematician seem to mostly fall into research, which I don't think I would mind, but would probably not be that great salary-wise. I know that you have talked about the possibility of going into the financial sector after you finish your PhD, I assume as some sort of analyst?
Although I still have ~6months to decide and a few (mostly CS with the exception of linear algebra) courses to attend before that, which I might find more interesting, I would still be interested in your thoughts about switching from CS to mathematics.
You've said that you got your bachelor's in environmental science and then switched to math for graduate school, but unfortunately for me, I won't be able to do this, since the CS bachelor's doesn't contain the necessary modules to pursue a master's in mathematics, so I would actually have to switch to a math bachelor's.

Either way I am interested to hear your opinion about this, and although the German and American system seem to be quite different, the general career paths should be the same.
Also sorry for the long comment :)

[termineitron]

Hi, I'm currently taking real analysis 1 in my undergraduate degree and I find it very useful to see some problems to know what to expect
Thank you for showing them

[kingmansee]

Never mind, I got the inequality sign wrong.
You can a weaker lower bound (said integral greater than 11) though.

[tanvec]

In the second problem, I believe you were wanting to say the Binomial Theorem (which of course is often represented by Pascal's Triangle)

[Korea-Study]

I'm always enjoying it. Thank you.

[omegads]

You're doing a Fantastic thing. If your solutions are available somewhere it would also be beneficial.