This is going to be the best Real analysis course on UA-cam. Thank You Professor! Your voice has reached all the way to the small islands of Maldives in Indian Ocean
Something felt very wrong at 10:10 . I tried it myself using the 'reverse calculation method we learnt earlier in the series and I obtained n > lnx/ln(1-e) Why can't we use that instead?
I feel that at 10:10, something is missing to ignore the "more" terms. Also, the second binomial term should have x^(n-1)/n not x^n/(n-1). at 15:40, I think it should be also noted that h converges to -1 on the [-1,0) interval because the exponent is 1 / odd number.
You're half-right :D The inequality does not necessarily go the other direction. What you want to say is rather "no we can’t, unless the inequality was on the other direction" A < B + C doe not necessarily mean that if you remove C from the inequality, A will become greater than B :) However, if you have A > B + C (and all terms are positive) then you can say with confidence that A > B
This video was helpful for getting some experience in using the definition of pointwise convergence but it does not convey an intuition or understanding of what it means for something to converge pointwise.
An interesting part of the last one is that the differentiability is only lost when you take the real root, rather than the principal root. Great video though, thanks!
We can formalize sequences a_n as functions a*: N -> R such that a_n = a*(n) by definition. Sequences of functions can then be formalized as function f*: (N x R) -> R. If we keep a given n fixed, we can extract the function f_n(x) = f*(x, n). If we keep a given x fixed, we can extract the sequence a_x(n) = f*(x, n). If every sequence a_x converges, we can define the limit function f(x) = lim a_x(n) as n -> oo. Note that the natural numbers have a single limiting behavior: going to infinity. In contrast, the real numbers have multiple limiting behaviors: going to plus/minus infinity and going to x for any x in R. The latter can be pointwise or uniform. We may want to study the interactions between the various kinds of limiting behavior. In either case, it will probably involve some combination of these elements: "for every epsilon > 0" and "there exists a natural N" and "there exists a delta > 0" and "for every n >= N" and "for every y with |x - y| < delta" and "the distance between f*(x, n) and either f*(y, n) or f(x) is less than epsilon" (and/or "consider a fixed x or n"). Reordering these for sequences might give "there exists a natural N such that for every epsilon > 0 and every n >= N we have |a_n - a| < epsilon". This says a_n is eventually constant (=a). Reordering the appropriate quantifies for pointwise continuity gives uniform continuity. We can study more reorderings: for every x there exists a delta > 0 such that for every epsilon > 0 and every y we have |x - y| < delta => |f(y) - f(x)| < epsilon. This says f is locally constant at every point x. If we put "for every x" below "there exists a delta" then the size of "locally" is bounded below by delta everywhere; adjacent delta-neighborhoods overlap, so by induction f is globally constant. Alright, that was fun. Now let's look at some convergence statements about sequences of functions. For example, for every x and every epsilon > 0 there exists delta > 0 and a natural N such that for all natural n and real y if |x - y| < delta and n >= N then |f*(x, n) - f*(y, n)| < epsilon. I would call this "eventual pointwise continuity": it's pointwise since the outermost quantifier is "for every x". It "eventual" since it holds for n >= N, not for every n, and it's a kind of continuity since the distance |some_f(x something) - some_f(y something)| can be made arbitrarily small by bringing x and y together. Note that this a limiting behavior in both the N-dimension and the R-dimension simultaneously. Looking at limiting behaviors one dimension at a time is often useful. For example, if every a_x converges to a limit function f which is continuous in x, then f is also pointwisely eventually continuous. [If |f(y) - f(x)| < epsilon for all n and y in (x-delta, x+delta) then this holds if n >= N.] Consider these function sequences: f*(x, n) = { h*(x, n-1) if n != 0 else 1 if x in Q else 0 } g*(x, n) = { h*(x, n) if x != 0 else (-1)^n } where h* is some function sequence with nice limiting behavior. The enterprising student may want to look at the sequence characterization of continuity and see whether some statement can be made and proven about the limiting behavior of (N x R) -> R objects in relation to the limiting behavior of (N x N) -> R objects.
Hey friend when you solved the hardest problem of geometry, it is a particular case of the movement of a function. That I developed. I leave it as data
This is going to be the best Real analysis course on UA-cam. Thank You Professor! Your voice has reached all the way to the small islands of Maldives in Indian Ocean
This channel is golden. So glad I found it. Great work Michael
Omg got my final exam tomorrow and this helps me so much!! I love you
Something felt very wrong at 10:10 . I tried it myself using the 'reverse calculation method we learnt earlier in the series and I obtained n > lnx/ln(1-e) Why can't we use that instead?
I feel that at 10:10, something is missing to ignore the "more" terms. Also, the second binomial term should have x^(n-1)/n not x^n/(n-1).
at 15:40, I think it should be also noted that h converges to -1 on the [-1,0) interval because the exponent is 1 / odd number.
4:37
lol
10:10 no we can’t, that inequality would go the other direction
You're half-right :D
The inequality does not necessarily go the other direction.
What you want to say is rather "no we can’t, unless the inequality was on the other direction"
A < B + C doe not necessarily mean that if you remove C from the inequality, A will become greater than B :)
However, if you have A > B + C (and all terms are positive) then you can say with confidence that A > B
Last two “ is all. Wts (e + ...)^n >1. So (e + ...)^n > x + nx^{(n-1)/n}e + ...
> x + nxe > x+1 > 1
You're by the best♥️♥️💙
It’s my favorite proof nice work bro
how can I have the intuition of Cal1 if I am doing that on my Calc1 exercises T.T
17:08
Real analysis for the win
This video was helpful for getting some experience in using the definition of pointwise convergence but it does not convey an intuition or understanding of what it means for something to converge pointwise.
An interesting part of the last one is that the differentiability is only lost when you take the real root, rather than the principal root.
Great video though, thanks!
Can someone explain the right way to prove the solution at 10:00?
It’s all there, just a few glitches. Want to show
(e + ...)^n > 1. So (e + ...)^n
> x + nx^{(n-1)/n}e + ...
> x + nxe
> x+1
> 1
At 6:24 I think you meant to write the interval [0,1)
No
Thank you!
You are good ❤️
@14:56 - @15:08 ??????
We can formalize sequences a_n as functions a*: N -> R such that a_n = a*(n) by definition.
Sequences of functions can then be formalized as function f*: (N x R) -> R.
If we keep a given n fixed, we can extract the function f_n(x) = f*(x, n).
If we keep a given x fixed, we can extract the sequence a_x(n) = f*(x, n).
If every sequence a_x converges, we can define the limit function f(x) = lim a_x(n) as n -> oo.
Note that the natural numbers have a single limiting behavior: going to infinity.
In contrast, the real numbers have multiple limiting behaviors: going to plus/minus infinity and going to x for any x in R. The latter can be pointwise or uniform.
We may want to study the interactions between the various kinds of limiting behavior.
In either case, it will probably involve some combination of these elements: "for every epsilon > 0" and "there exists a natural N" and "there exists a delta > 0" and "for every n >= N" and "for every y with |x - y| < delta" and "the distance between f*(x, n) and either f*(y, n) or f(x) is less than epsilon" (and/or "consider a fixed x or n").
Reordering these for sequences might give "there exists a natural N such that for every epsilon > 0 and every n >= N we have |a_n - a| < epsilon". This says a_n is eventually constant (=a).
Reordering the appropriate quantifies for pointwise continuity gives uniform continuity. We can study more reorderings: for every x there exists a delta > 0 such that for every epsilon > 0 and every y we have |x - y| < delta => |f(y) - f(x)| < epsilon. This says f is locally constant at every point x. If we put "for every x" below "there exists a delta" then the size of "locally" is bounded below by delta everywhere; adjacent delta-neighborhoods overlap, so by induction f is globally constant.
Alright, that was fun. Now let's look at some convergence statements about sequences of functions. For example, for every x and every epsilon > 0 there exists delta > 0 and a natural N such that for all natural n and real y if |x - y| < delta and n >= N then |f*(x, n) - f*(y, n)| < epsilon.
I would call this "eventual pointwise continuity": it's pointwise since the outermost quantifier is "for every x". It "eventual" since it holds for n >= N, not for every n, and it's a kind of continuity since the distance |some_f(x something) - some_f(y something)| can be made arbitrarily small by bringing x and y together.
Note that this a limiting behavior in both the N-dimension and the R-dimension simultaneously.
Looking at limiting behaviors one dimension at a time is often useful. For example, if every a_x converges to a limit function f which is continuous in x, then f is also pointwisely eventually continuous. [If |f(y) - f(x)| < epsilon for all n and y in (x-delta, x+delta) then this holds if n >= N.]
Consider these function sequences:
f*(x, n) = { h*(x, n-1) if n != 0 else 1 if x in Q else 0 }
g*(x, n) = { h*(x, n) if x != 0 else (-1)^n }
where h* is some function sequence with nice limiting behavior.
The enterprising student may want to look at the sequence characterization of continuity and see whether some statement can be made and proven about the limiting behavior of (N x R) -> R objects in relation to the limiting behavior of (N x N) -> R objects.
number 1 ❤
Hey friend when you solved the hardest problem of geometry, it is a particular case of the movement of a function. That I developed. I leave it as data
Is the nth root of a number not imaginary sorry for asking this stupid question 😅
Real nth roots exist when n is odd when n is even all the roots are complex (with no real roots)
gg m8
god