8:30 seems a bit too complicated: we assumed that r is rational and thus b^r is an element of B(r). Hence, sup B(r) must be at least b^r by the (usual) definition of suprema as (least) upper bounds. Again, equality follows as b^r also is an upper bound on B(r), as in your proof.
@JoelRosenfeld
Жыл бұрын
Right, I thought about that, and certainly that works for rationals. But you don’t have equality for irrationals. I think Rudin might have meant to have the problem without the less than or equal qualifier, so I went the extra mile and gave a proof that would work for that too.
@lemonflavouredtnt6969
Жыл бұрын
Good video prof. I did 2 analysis modules and some of this stuff we never touched on at all. E.g. we always assumed the well-definedness of exponentiation. Makes me worry what I’ve missed!
@Cyclonus-fc1xx
Жыл бұрын
That advice to verify every statement in the proof and not just if it proves the original statement is really helpful, I really enjoy this kind of video where you get to learn how to think about a proof. By the way, nice waffle and coffee now I want one aswell
@JoelRosenfeld
Жыл бұрын
I’m glad you like the video! Part of why I’m making these is because I wish I had something like this when I was taking the class 16 years ago. Lol I recently discovered stroopwafels, they have become a favorite snack
@Cyclonus-fc1xx
Жыл бұрын
@@JoelRosenfeld Will definitely try them out if I ever come across them made them look too tempting
@jwiskikruger8921
Жыл бұрын
i love your channel, you're so inspiring!
@JoelRosenfeld
Жыл бұрын
Thank you so much! I'm happy to have you here.
@darkpikachu_.
Жыл бұрын
really cool video again😊
@JoelRosenfeld
Жыл бұрын
Thank you! I'm glad you liked it!
@crimedogs
Жыл бұрын
Holy Moly. . . I need to go back to your first video! You are one clever dude!
@JoelRosenfeld
Жыл бұрын
Thank you! I’m glad you like it!
@mmariokart231
Жыл бұрын
I’ve got most of this solved but I’m stuck on the second to last paer
@JoelRosenfeld
Жыл бұрын
What has you stuck in particular? This is a tough problem.
@mmariokart231
Жыл бұрын
@@JoelRosenfeld the step right before we have to show x is unique
@JoelRosenfeld
Жыл бұрын
Ok, I’ll take a look this weekend and see if I can help clear it up some more
@ejrobb81
Жыл бұрын
This brought back some baby rudin nightmares
@JoelRosenfeld
Жыл бұрын
lol I didn’t realize I was running a Horror KZitem Channel!
@ejrobb81
Жыл бұрын
I think most first encounters with Rudin are...shocking...super slick proofs in the chapter make it seem manageable and then you get to the problem sets ...this is where the true battle begins or the sleep deprivation.
@JoelRosenfeld
Жыл бұрын
Very true. Rudin is a little too slick with his proofs. It takes a lot to parse through them to see what he’s actually doing. One reason I’m making this series is to help my students understand Real Analysis from both Rudin and other perspectives. It’s a fun project to dissect Baby Rudin.
@ejrobb81
Жыл бұрын
@@JoelRosenfeld It is strange with Rudin his texts do actually get better when you go back through them after measure theory and functional analysis then I think one can appreciate them more and have fun with them maybe it is the lack of a midterm or final involved or the fear of how many points you are going to lose on the problem set (assuming you miracle your way to answering them all ;) ) or just having the proverbial mathematical maturity. In the end he is a great way to find out if you really do love math!
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