Real Analysis, Lecture 6: Principle of Induction

[@peki_ooooooo]

"You can think of it as a proof technique, but it really you should think of it as an axiomatic property of the natural numbers." Really enlightment.

[@jarrodanderson2124]

I count myself incredibly lucky to have had a brilliant analysis professor, just like Professor Su.

[@luisribeiro9407]

really great lecture, he's really a good teacher that seams to be very knowledgeable and concise in the subject. Thank you for sharing with us.

[@PsychonautAtom]

At my university, Real Analysis is taken after 3 semesters of Calculus, after a semester of discrete mathematics, and after a semester of Advanced Calculus (Into. to analysis).

[@Myrslokstok]

Yes the resolution is not good. But the teacher has a nice voice and is clear in his mind, the sound is good. You can understand what is written, and you see the board from above witch I like.

[@alfiealfie35]

which not witch but yeah i agree

[@milksushi6640]

As a naive child, I never thought a horse (or rather, an arbitrary set of horses) would violate the depths of my brain. I was wrong.

[@sdoken]

i dont understand why they could not increase the resolution. these lectures are great but the board is hard to read because of low resolution!

[@ShokoDemon]

in my university it's called Analysis and is a canonical 1st semester course for mathematics students. it might be taught as calculus in other universities.

[@michaelsohnen6526]

This video is so old that i don't know if anyone will see this comment, but I can point two problems with one of the inductive proofs. 1: in the tiling case, his argument makes sense, but it requires that a 2^n be tileable in such a way that the removal is specifically in one of the corners, not in the interior. He does not address that problem, as the task he proposed when starting the proof allows for the removal of any interior piece, including but not limited to corner pieces. 2: Though this is technically not a logical issue, it is a mechanical issue. The proof for the tiling problem only shows that a grid can be tiled in a highly organized pattern with a sort of expanding/radial symmetry. It does not show that every 2^n board can be tiled chaotically.

[@holomorphy]

The 2^(n+1) board is divided into 4 2^n 'corner' boards, with no interior. If you mean that the a single tile is removed only from a corner of a 2^n board you are mistaken (and he doesn't even draw it that way). The proof shows that if you have any 2^k by 2^k grid (where 2 is a positive integer) and you remove *any* single piece, you can tile the remainder using the described L-shaped tiles. No 'highly organized pattern' is given. Do you disagree?

[@michaelsohnen6526]

Jackson Knox I'll have to watch it again to see for sure, thanks for the reply!

[@rampadmanabhan4258]

I think this is essentially a proof that every number of the form 2²ⁿ is divisible by 3? I understood the chessboard analogy, but this just seemed a little simpler to me.

[@SequinBrain]

The three interior squares are the corners of the remaining 3 4x4 tile sets. So, it is in one of the 4 necessary corners. For your #2, some of the necessary conditions like that are left out, and one of his drawings did show that you can't just randomly pick an area. I find this sometimes in proofs, they forget to define this or that or leave out a restriction that we can assume is necessary, till 10 years pass and we forget what that restriction actually was.

[@bentupper4614]

You just can't assume you have three horses like you used to.

[@Myrslokstok]

Interesting example of induction - good teacher. Have this subject "Real analyst" other names as well - can't se that my university have this courses.

[@kiyeoblee3812]

fairly good lectures, but whenever he writes on the board, I need to playback again and again either to listen to what he writes or to guess his hand motion

[@yash9shhreya]

Professor is awesome !!

[@dopplerdog6817]

When I grow up I want to be a real analyst

[@Myrslokstok]

Thanks!

[@NewCalculus]

The principle of induction was known since Classical Greece. It has nothing to do with real analysis - the garbage that is all about an object that does not exist - "real number".

[@HabibuMukhandi]

The hypotenuse of a triangle whose two sides are of unit length does not exist?

[@darksecret965]

lol imagine this guy's surprise when he finds out about complex numbers, quaternions, dual numbers, p-adic numbers, ordinals

[@adelaidekhayon2631]

if we take 2^0 x 2^0 the example would be false

[@enesalbay9707]

You can start from n=0 but in this case if you remove a sqaure, there will be no square that needs to be filled. Therefore, it is trivial case.

[@SequinBrain]

if we're using n as an element of the natural numbers (2^n), 0 isn't an element. 0 is a whole number, not a natural one, probably for the reason you mentioned.

[@peki_ooooooo]

I get at here.