Exactly the way I proved it as well. I just said that, without loss of generality, we can assume c = 0 (we can consider g(z) := f(z) - c which is continuous, so if we show that g(z) = 0 for a z in (a,b), we know that f(z) = c for this exact z) which makes everything a bit easier to write down and visualize.
The set I= ]-inf;a[ is not closed in R, which means, there exist sequence of element of I, which converges, but the limit is no longer in I The sequence s_n is exactly this kind of sequence The same goes for your (1/n) sequence Each element of your sequence is inside ]0;+inf[ but your limit is not in ]0;+inf[
yes, this proof is beautiful. But would you agree "beauty is in the eye of the beholder" so we can not all be viewed beautiful even though we are. How do you feel about route to the summit using the nested interval theorem.
Did you ever give a topological version of this proof, not using metric spaces but proving that continuous image of connected is connected? There's all these topological properties that continuous functions preserve, and you have to remember whether it's f or f^-1, like open, closed, compact, connected, etc. And each of them is a new theorem/definition when interpreted in the metric space context.
Great vid. We can also use open balls instead of sequences. Suppose f(x0) < c. Then by continuity of f there is an open ball about x0 whose image is < c. This ball contains points in S greater than x0 which contradicts the fact that x0 is an upper bound of S. On the other hand, suppose f(x0) > c. Then by continuity of f there is an open ball about x0 whose image is > c. This ball contains upper bounds of S less than x0 which contradicts the fact the x0 is the _least_ upper bound of S. Hence f(x0) = c.
Speaking of real analysis, I looked at Folland's 'Real Analysis' the other day. I found two of his definitions extremely interesting: the limit supremum and limit infimum of a sequence of _sets_ E1, E2, E3, ... One is defined as the intersection from k=1 to oo of the union from i=k to oo of Ei. The other is the same but with "union" and "intersection" swapped. To make sure I understood, I intentionally forgot which was which and tasked myself with figuring it out based on what makes conceptual sense. These definitions suggest that an arbitrary sequence of sets "converges" if its limit supremum equals its limit infimum. I wonder how this relates to traditional limits in calculus and topology.
@@drpeyam Hi Dr. Peyam, I checked out the playlist but I only found content for sequences of real numbers, not sequences of arbitrary sets. Maybe I just missed it. I can see if you take real numbers to be Dedekind cuts then the notions are actually equivalent.
Can the IVT be restated as the following? If f is continuous on the interval [a,b], then for every c between f(a) and f(b), c belongs to the image set of f([a,b]).
Can you do a video on line integrals which are very generalized I get it for like R¹ or R² but it's hard to visualize for R^n/ n-integrals like there are double and triple integrals I want to know (and think something interesting will happen in the infinite case but I maybe wrong)
Are you using Pugh's Real Mathematical Analysis book for this course? I'm using it for a course right now and a lot of the examples you give are the same.
Here is a somehow related questions, about upper bounds for maximum slew rate of functions.. I hope you could talk about it on your videos: on math stack exchange, question 4269062
I've been wanting a solid proof of IVT and here it is. Thank you!
Wait . . .till now I thought that the IVT was intuitive. I mean it is. But there exists a proof ? And that too formal ? Whoa ! COOL !
oh my god we have the same profile picture
yay !@@damyankorena
Exactly the way I proved it as well. I just said that, without loss of generality, we can assume c = 0 (we can consider g(z) := f(z) - c which is continuous, so if we show that g(z) = 0 for a z in (a,b), we know that f(z) = c for this exact z) which makes everything a bit easier to write down and visualize.
Hello there! My school textbook has a proof of IVT by using Bolzano Theorem which states if a function is continuous at [a, b] and f(a)*f(b)
Same thing!
sir i love you so much, you're one of the few UA-camrs that I think are wonderful people in real life too
Thank you!!
Sir, very informative
12:15 Did you mean for N "very big" instead of "very small"?
I haven't seen the proof in a long time. Thanks 🍉❤❤🌷
Thank you so much for making these videos. They have really helped my university maths as everything you explain makes so much sense!
Thank you Dr Peyam. Always is a pleasure watch you channel.
How come at 10:20 the limit is
Wait no never mind, disregard that ^^ i convinced myself haha say: (-1/n) < 0 but lim (-1/n) = 0
Yep :)
The set I= ]-inf;a[ is not closed in R, which means, there exist sequence of element of I, which converges, but the limit is no longer in I
The sequence s_n is exactly this kind of sequence
The same goes for your (1/n) sequence
Each element of your sequence is inside ]0;+inf[ but your limit is not in ]0;+inf[
Your analysis proofs are always beautiful
yes, this proof is beautiful. But would you agree "beauty is in the eye of the beholder" so we can not all be viewed beautiful even though we are. How do you feel about route to the summit using the nested interval theorem.
Did you ever give a topological version of this proof, not using metric spaces but proving that continuous image of connected is connected? There's all these topological properties that continuous functions preserve, and you have to remember whether it's f or f^-1, like open, closed, compact, connected, etc. And each of them is a new theorem/definition when interpreted in the metric space context.
Wow!! Elegantly explained Boss 🤩
Thanks for posting the video ❤🌿
You’re welcome!!!
@@drpeyam ❤❤🌿🌿
The proof is absolutely 😘 indeed
Hello Sir, could you please elaborate a bit more on how you used
1-1/n < 1 for all n but the limit is 1
@@drpeyam Thank you Sir.
well done my friend
Great vid. We can also use open balls instead of sequences. Suppose f(x0) < c. Then by continuity of f there is an open ball about x0 whose image is < c. This ball contains points in S greater than x0 which contradicts the fact that x0 is an upper bound of S. On the other hand, suppose f(x0) > c. Then by continuity of f there is an open ball about x0 whose image is > c. This ball contains upper bounds of S less than x0 which contradicts the fact the x0 is the _least_ upper bound of S. Hence f(x0) = c.
Speaking of real analysis, I looked at Folland's 'Real Analysis' the other day. I found two of his definitions extremely interesting: the limit supremum and limit infimum of a sequence of _sets_ E1, E2, E3, ...
One is defined as the intersection from k=1 to oo of the union from i=k to oo of Ei. The other is the same but with "union" and "intersection" swapped. To make sure I understood, I intentionally forgot which was which and tasked myself with figuring it out based on what makes conceptual sense. These definitions suggest that an arbitrary sequence of sets "converges" if its limit supremum equals its limit infimum. I wonder how this relates to traditional limits in calculus and topology.
There are videos precisely on that, check out my sequences playlist
@@drpeyam Hi Dr. Peyam, I checked out the playlist but I only found content for sequences of real numbers, not sequences of arbitrary sets. Maybe I just missed it. I can see if you take real numbers to be Dedekind cuts then the notions are actually equivalent.
Thank you very much.
Hi Dr Peyam at 14:18 why f of TN is greater than or equal c ？I think it's just greater than c only no equal sign.
Thanks.
Yes but if it’s > c then it’s >= c
Can you prove in a video that the area of an squircle x^n+y^n=1 es equal to 4*(gamma(1+1/n))^2/gamma(1+2/n)?
That’s a nice idea
Can the IVT be restated as the following? If f is continuous on the interval [a,b], then for every c between f(a) and f(b), c belongs to the image set of f([a,b]).
Yes, by definition of image set
Thank you! I c what you did there!
Baby analysis : Proof of IVT
Adult analysis: Proof of IFT 😂
Can you do a video on line integrals which are very generalized I get it for like R¹ or R² but it's hard to visualize for R^n/ n-integrals like there are double and triple integrals I want to know (and think something interesting will happen in the infinite case but I maybe wrong)
Nice prrof. What about cool integral are you planning one?
Are you using Pugh's Real Mathematical Analysis book for this course? I'm using it for a course right now and a lot of the examples you give are the same.
I love that book!!! Lucky you, it was the same I used when I took analysis. But no, the videos are based on the book by Ross
Here is a somehow related questions, about upper bounds for maximum slew rate of functions.. I hope you could talk about it on your videos: on math stack exchange, question 4269062
중간값의 정리인가요?
In english it's called
Intermediate Value Theorem
Hope this helps :D
Always radioactive💫🤍
中值定理
Back to classroom????
No that video was from May 2020
@@drpeyam oh
@@chessematics xD