Completeness

Completeness

In this video, I define the notion of a complete metric space and show that the real numbers are complete. This is a nice application of Cauchy sequences and has deep consequences in topology and analysis

[blackpenredpen]

No wonder I never feel my life is completed...

[NotoriousSRG]

I love that you never pronounce “Cauchy” the same way twice lol

[vadimpavlov]

I think the final step relies on the fact that R contains all its lim sups and lim infs, for otherwise the same argument would apply to Q

[arijeetsingh]

This channel probably has the lowest dislike/like ratio! Way to go Dr. Peyam.

[hauntedmasc]

This video on completeness is a (Dedekind) cut above the rest!

[toaj]

Can you please explain where this proof breaks down if our range is restricted to Q? Is it that the Limsup and Liminf do not exist?

[dmytro_shum]

Sequence: sum 1/k (harmonic series) for k from 1 to n is not bounded. Is it a Cauchy sequence?

[toaj]

A space can be incomplete if we can get an element outside that space in the limit. Is there any other way in which spaces can be incomplete?

[pocojoyo]

Dr Peyam, is this part of your Elementary Real Analysis course ? 1 or 2 ?

[fynman]

Where was this video when I took my Analysis Exam one month ago!?

Still a good video though! 👍

[coreymonsta]

it's not surprising after you see it's just about technicalities regarding where elements lie

[shiina_mahiru_]

I have already forgotten the entire proof from my analysis class 😂

[gabrielmendozallanes]

UC Irvine is not complete without Dr. Peyam

[Galileopi]

Brilliant as always, mate, just on my birthday, thanks

[henrikfischbeck]

ure saying supremum and max as if they where the same. Also u say Q has holes, but then u say Z is complete...

if it was lim min = lim max i would undetrstand but..

Uhm?...

Imagine a sequence of rationals getting closer and closer to sqrt2... the two numbers next to sqrt2 are rational ( because between two rationals is an irrational. and vice versa) so a sequence of rationals alternating around sqrt2 getting closer and closer have liminf(q)=sqrt(2)=limsup(q) ? like Q is dense in R so the smallest distance d(q, sqr2). but not if we change supp and inf for min and max..?

[jkid]

You're not building up to p-adics or something, are you? :)