Real Analysis | Sequences of functions and integration.

We prove that if a sequence of (Riemann) integrable functions converges uniformly, then its limit is also integrable. Further, we show that the limit may not be brought inside the integral if convergence is pointwise.
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[@tomatrix7525]

Loving this series. I’m not actually taking this class but it’s a helpful addition of proofs etc to the math ‘repertoire’ I have

[@sebastianferra6288]

Such a great channel, like your style and the clarity of your voice, its fun to watch math while eating or on a bad day, or any at any time really! Keep up the good work! Greetings from Chile

[@MCMCFan1]

What happened with the rest of the proof after 8:07?

[@JSSTyger]

00:00 And thats a good place to start.

[@s4623]

3:23 this reminds me that it would be a good idea to have a lecture series which teaches people how to figure that out ahead of time. Just like all the other "How did you come up with that?" moments.

[@tomatrix7525]

10:40 error, shouldn’t it be (-n(1-x^2))/2 Edit: Sorry he is correct, he just forgot the nth power in the above line so it mislead me.

[@TheMauror22]

Hey but you didn't prove that you can "pass de limit inside the integral", did you? You just prove that f is integrable.

[@svsharma3893]

Thank You Michael

[@chongweifongchongweifong7180]

Where is the proof for the statement at 7:56?

[@morbidmanatee5550]

Why isn''t it (1-x^2)^n in the last integral? Shouldn't that be 2nx(1-x^2)^n ?

[@goodplacetostop2973]

11:33

[@djvalentedochp]

hey michael nice job

[@s4623]

9:56 why the nth power was missing from the integral of fn?

[@megacarlosloki]

Could you see the first question of "provaextramuros" of 2018 plz?

[@andreypillay4323]

Hey there, I may be applying to Randolph this year, any tips?

[@indarajgochermaths5176]

Nice sir

[@joaopedrobmenezes2977]

Problem sugestion! ( I know I already sent it in the last video, but I hope you will see this time!): "(Brazillian undergraduate olympiad 2019 ) Let R> 0 be the set of positive real numbers. Determine all functions f: R> 0 → R> 0 such that f (xy + f (x)) = f (f (x) f (y)) + x for all positive reals x and y.

[@ryszcl1673]

Does that mean x and y can't equal 0?

[@joaopedrobmenezes2977]

@@ryszcl1673 yep

[@djvalentedochp]

já sugeri esse problema tbm, é mt bom

[@CM63_France]

Hi, For fun: 1 "so let's may be go ahead and do that", 1 "so let's go head and", 1 "now I'll go ahead and", 1 "ok, great", 11:21 : "one is not equal to zero".

[@poi_aithhkunnnRVC]

11:33 good place to stop first

[@JamesMonteroso]

he looks like paul dirac

[@GhostyOcean]

Looks like I really messed up my Analysis final exam by skipping uniform convergence 👉😅👉

[@GhostyOcean]

Also, you forgot to raise (1-x²) to the nth power in the integral, but kept the power rule.