the subtleties of sequences of functions

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[edmundwoolliams]

16:49 I love the "infinity is not equal to zero" part

[ed.puckett]

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.

[aqeel]

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.

[Buridan]

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.

[matheusjahnke]

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).

[atlas]

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

[barutjeh]

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.

[thatspookyfeeling]

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?

[theelk]

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.

[oliviermiakinen]

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.

[aweebthatlovesmath]

But does the function fn uniformly converge if we restrict it
to [0, 1)?

[ingiford]

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)

[charleyhoward]

17:34 WAS a good place to stop ! Whew !

[mrminer]

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!"

[naffouri]

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)

[minwithoutintroduction]

كنت أحتاج إلى هذا الفيديو في الموسم الجامعي 1994/1995

[icewlf]

an example that converges pointwise but not uniformly on entire R: f_n(x)=sin(arctan(nx))

[Alan-zftt]

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?

[plumbr]

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.

[gp-htug]

What a confusing video. I had to watch it more then once