Wow, that is subtle and useful. It's a lot to think about. Especially that g_n' does not converge to g' even when g_n converged to g. Thank you once again for a great video.
For the first part, we can also notice that x^n takes on all values between 0 and 1 inclusive on the interval [0, 1] since it is strictly increasing and 0^n = 0, 1^n = 1 for any value of n. Then, setting x^n = a, x^2n must equal a^2. a-a^2 = -(a-0.5)^2 + 1/4. Hence, we obtain a maximum value of 1/4 when a=0.5 (or x = (2)^{1/n}). No need to use differentiation! Just basic algebra.
it would help if the name of the video would state something like pointwise or uniformely convergence. If searching for "that one video" where he gives a good example for a comparison of them, it would then be easier to find.
Other example of point-wise convergence but not necessarily uniform convergence is Fourier Series when the function has discontinuities; The discontinuities, when approximated by a Fourier series, are translated into a "bump" which overshoots and, just like fn(x)=x^n - x^(2n) example, as you increase the number of terms... the bump gets thinner and thinner(in the x axis)... but still always over a certain depth(y axis).
You should do a follow-up video about Dini's theorem. On a closed and bounded set, monotone, pointwise convergence implies uniform convergence. You can give nice counter examples why you can't leave out any of those requirements.
I say he should also do a follow-up on the other ways to show uniform convergence. Personally I like the way of this theorem: IF f_n converges pointwise to f on a set D and let S_n to be the supremum of |f_n(x)-f(x)|, where x is in D, THEN f_n converges uniformly to f on D if and only if S_n goes to 0 as n goes to positive infinity.
@@seanbastian4614 Easy to prove. First WLOG f = 0 by replacing f_n with f_n - f. Then S_n -> 0 iff for any given e>0 eventually S_n < e iff e satisfied for uniform definition iff f_n -> 0 uniformly. But I bet he'd stretch it out into a 10 minute video.
@@aadfg0 i know how to prove it. I don’t think others know about that and I feel it’s worth people knowing about if they’re having to deal with convergence of sequences of functions.
Dini's Theorem also requires that the point-wise limit function be continuous, and that's also a necessary restriction for the theorem to hold. Counter-example without that restriction: f_n : [0,1] --> [0,1] by f_n(x) = x^n. Dini's Theorem also requires that each function f_n be continuous.
I find it helpful to think of uniform convergence in terms of the sup metric, so f_n → f only if the largest difference between f_n and f goes to 0 as n increases
11:09 this is not exactly the negation of the previous définition. For example take f_n(x) = x^n - x^(2n) for even values of n but f_n(x) = 0 for odd values. It does not satisfy the "negated" condition but it also does not satisfy the original one.
For anyone wondering what the difference is, his "negation" is ignoring the if n>=N part of the uniform convergence definition. When proving the negation, for the existing epsilon, the condition doesn't have to fail for every single N, but for any particular N there must be some bigger n for which it does fail.
@@lynnjones5587 Not quite. For any N, you need it to fail for some bigger n. So if your one contradictory case happened at n=1000, then for N=1001 there wouldn't be a failure for any n>N. This in fact means you need an infinite number of failing n's.
I remember asking my real analysis teacher if it was possible for a sequence of continuous functions to converge pointwise but not uniformly to a continuous function and this was exactly the example he gave. I don’t remember if I asked if compactness of the domain mattered but that obviously is covered here as well.
If you have time and inclination, would appreciate your explanation of how Cantor came up with the whole idea of accumulation points. I watched Walter Rudin's video lecture but could not hear it well enough for some reason. And if possible, maybe comparison of the relative advantages of Dedekind Cuts vs Cauchy Sequences, is there anything that unifies them in a general setting?
The way I think about it which makes it very easy is that uniform convergence is the same as convergence in the sup norm. Since the function is continuous, adding or removing finitely many points doesn't change the supremum.
Regardless the answer, that's the right kind of question to be asking! Studying higher math is significantly the process of asking such questions, and then trying to answer them for yourself with either a proof or a counter-example.
Think about x^n on [0,1]. The value at 0 is 0. The value at 1 is 1. It makes a curve between the two that starts near 0, then has a smooth "corner" and flies up to 1. This will converge to 0 for 0≤x
Reminds me of the function of equations as n in N, of f(x) = (1- (x^ (2n) )) ^ (1 / 2n) Which goes from a half circle circle at N = 1 to a half square as you go to infinity. ((I hope i got the function right from memory)
Incredible! Such a sweet looking relation in function form with xϵ[0,1], xⁿ and x²ⁿ with nϵℕ naturally And if that was not contorted enough add in fₙ(x) depending on xϵ[0,1] AND nϵℕ naturally AND xⁿ and x²ⁿ. If we add in two versions of convergence, pointwise and uniform along with convergence of xⁿ and x²ⁿ and convergence of functions ≡ shrug?
Do you mean why are we picking (1/2)^(1/n)? It's the peak of the hump, the x value that always gives a fixed maximum of 1/4 for the function f_n. It's the most egregious, and demonstrates with the most clarity what we're referring to when we ask the whole function to converge uniformity (or fail to do so).
To be fair, he’s using a definition of the natural numbers that excludes 0 and what he wrote is true for all strictly positive integers n. It’s only resulting in 1=0 when n=0. But yeah, whether or not 0 is a natural number depends on the context of who’s using it. (This channel does a lot of number theory and in that context 0 is usually excluded, but if you’re using a set theoretic version of the Naturals where they are defined as all possible cardinalities of finite sets then 0 is included since the Empty set has cardinality 0.)
16:49 I love the "infinity is not equal to zero" part
Wow, that is subtle and useful. It's a lot to think about. Especially that g_n' does not converge to g' even when g_n converged to g. Thank you once again for a great video.
For the first part, we can also notice that x^n takes on all values between 0 and 1 inclusive on the interval [0, 1] since it is strictly increasing and 0^n = 0, 1^n = 1 for any value of n. Then, setting x^n = a, x^2n must equal a^2.
a-a^2 = -(a-0.5)^2 + 1/4. Hence, we obtain a maximum value of 1/4 when a=0.5 (or x = (2)^{1/n}).
No need to use differentiation! Just basic algebra.
it would help if the name of the video would state something like pointwise or uniformely convergence. If searching for "that one video" where he gives a good example for a comparison of them, it would then be easier to find.
Perhaps he was worried that title might intimidate people and they wouldn't click?
Other example of point-wise convergence but not necessarily uniform convergence is Fourier Series when the function has discontinuities;
The discontinuities, when approximated by a Fourier series, are translated into a "bump" which overshoots and, just like fn(x)=x^n - x^(2n) example, as you increase the number of terms... the bump gets thinner and thinner(in the x axis)... but still always over a certain depth(y axis).
You should do a follow-up video about Dini's theorem. On a closed and bounded set, monotone, pointwise convergence implies uniform convergence. You can give nice counter examples why you can't leave out any of those requirements.
I say he should also do a follow-up on the other ways to show uniform convergence. Personally I like the way of this theorem: IF f_n converges pointwise to f on a set D and let S_n to be the supremum of |f_n(x)-f(x)|, where x is in D, THEN f_n converges uniformly to f on D if and only if S_n goes to 0 as n goes to positive infinity.
Perhaps a follow-up on quasi-uniform convergence and Arzela's theorem would also be of interest.
@@seanbastian4614 Easy to prove. First WLOG f = 0 by replacing f_n with f_n - f. Then S_n -> 0 iff for any given e>0 eventually S_n < e iff e satisfied for uniform definition iff f_n -> 0 uniformly. But I bet he'd stretch it out into a 10 minute video.
@@aadfg0 i know how to prove it. I don’t think others know about that and I feel it’s worth people knowing about if they’re having to deal with convergence of sequences of functions.
Dini's Theorem also requires that the point-wise limit function be continuous, and that's also a necessary restriction for the theorem to hold.
Counter-example without that restriction:
f_n : [0,1] --> [0,1] by f_n(x) = x^n.
Dini's Theorem also requires that each function f_n be continuous.
I find it helpful to think of uniform convergence in terms of the sup metric, so f_n → f only if the largest difference between f_n and f goes to 0 as n increases
17:34 WAS a good place to stop ! Whew !
11:09 this is not exactly the negation of the previous définition. For example take f_n(x) = x^n - x^(2n) for even values of n but f_n(x) = 0 for odd values. It does not satisfy the "negated" condition but it also does not satisfy the original one.
For anyone wondering what the difference is, his "negation" is ignoring the if n>=N part of the uniform convergence definition. When proving the negation, for the existing epsilon, the condition doesn't have to fail for every single N, but for any particular N there must be some bigger n for which it does fail.
@@burk314 So, as in, one contradictory case, proves the entire theory false?
@@lynnjones5587 Not quite. For any N, you need it to fail for some bigger n. So if your one contradictory case happened at n=1000, then for N=1001 there wouldn't be a failure for any n>N. This in fact means you need an infinite number of failing n's.
I remember asking my real analysis teacher if it was possible for a sequence of continuous functions to converge pointwise but not uniformly to a continuous function and this was exactly the example he gave. I don’t remember if I asked if compactness of the domain mattered but that obviously is covered here as well.
If you have time and inclination, would appreciate your explanation of how Cantor came up with the whole idea of accumulation points. I watched Walter Rudin's video lecture but could not hear it well enough for some reason. And if possible, maybe comparison of the relative advantages of Dedekind Cuts vs Cauchy Sequences, is there anything that unifies them in a general setting?
But does the function fn uniformly converge if we restrict it
to [0,1)?
No, adding or removing can't change uniform convergence. Although if you restrict to [0,a] for a fixed a
The way I think about it which makes it very easy is that uniform convergence is the same as convergence in the sup norm. Since the function is continuous, adding or removing finitely many points doesn't change the supremum.
Regardless the answer, that's the right kind of question to be asking!
Studying higher math is significantly the process of asking such questions, and then trying to answer them for yourself with either a proof or a counter-example.
Think about x^n on [0,1]. The value at 0 is 0. The value at 1 is 1. It makes a curve between the two that starts near 0, then has a smooth "corner" and flies up to 1.
This will converge to 0 for 0≤x
Oh man! The memories of my UCSD math degree! I saw that perma-spike in the animation, and thought "Uh-Oh, uniform continuity NOT GONNA HAPPEN!"
Reminds me of the function of equations as n in N, of f(x) = (1- (x^ (2n) )) ^ (1 / 2n) Which goes from a half circle circle at N = 1 to a half square as you go to infinity. ((I hope i got the function right from memory)
كنت أحتاج إلى هذا الفيديو في الموسم الجامعي 1994/1995
Great example
Incredible! Such a sweet looking relation in function form with xϵ[0,1], xⁿ and x²ⁿ with nϵℕ naturally
And if that was not contorted enough add in fₙ(x) depending on xϵ[0,1] AND nϵℕ naturally AND xⁿ and x²ⁿ.
If we add in two versions of convergence, pointwise and uniform along with
convergence of xⁿ and x²ⁿ and convergence of functions ≡ shrug?
an example that converges pointwise but not uniformly on entire R: f_n(x)=sin(arctan(nx))
Good point. The target function is discontinuous at 0, and each f_n is continuous, so the convergence cannot be uniform.
Why is 1/n-sqrt(2) special? The same can be applied to any 1/n-sqrt (like 1/n-sqrt(3) has a value of 2/9 regardless of n)
Do you mean why are we picking (1/2)^(1/n)? It's the peak of the hump, the x value that always gives a fixed maximum of 1/4 for the function f_n. It's the most egregious, and demonstrates with the most clarity what we're referring to when we ask the whole function to converge uniformity (or fail to do so).
At just a minute in, we're told that one minus zero is zero. It's a long time since I got my degree, but one minus zero used to be one.
He meant one minus one
So much for constructive criticism
To be fair, he’s using a definition of the natural numbers that excludes 0 and what he wrote is true for all strictly positive integers n. It’s only resulting in 1=0 when n=0.
But yeah, whether or not 0 is a natural number depends on the context of who’s using it. (This channel does a lot of number theory and in that context 0 is usually excluded, but if you’re using a set theoretic version of the Naturals where they are defined as all possible cardinalities of finite sets then 0 is included since the Empty set has cardinality 0.)
We found him, the guy who never makes mistakes 😳😳😳
It's that "new math" everyone's been talking about. Oh, wait. That was in the 1960s. Never mind. It's been a long time since I got my degree, too.
What a confusing video. I had to watch it more then once