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.
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.
It should be noted at 6:49 that if exactly one of the denominators s2, t2 is negative then it does not hold that s < t implies s1t2 < s2t1. However, the lemma still holds because if exactly one of the denominators is negative, we can take the -(s2t2) root of the inequality b^(-(s1t2)) < b^(-(s2t1)), and the rest follows.
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!
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
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
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.
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.
@@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!
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.
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.
It should be noted at 6:49 that if exactly one of the denominators s2, t2 is negative then it does not hold that s < t implies s1t2 < s2t1. However, the lemma still holds because if exactly one of the denominators is negative, we can take the -(s2t2) root of the inequality b^(-(s1t2)) < b^(-(s2t1)), and the rest follows.
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!
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
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
@@JoelRosenfeld Will definitely try them out if I ever come across them made them look too tempting
i love your channel, you're so inspiring!
Thank you so much! I'm happy to have you here.
really cool video again😊
Thank you! I'm glad you liked it!
This brought back some baby rudin nightmares
lol I didn’t realize I was running a Horror TH-cam Channel!
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.
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.
@@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!
I’ve got most of this solved but I’m stuck on the second to last paer
What has you stuck in particular? This is a tough problem.
@@JoelRosenfeld the step right before we have to show x is unique
Ok, I’ll take a look this weekend and see if I can help clear it up some more
Holy Moly. . . I need to go back to your first video! You are one clever dude!
Thank you! I’m glad you like it!