Intro to Cauchy Sequences and Cauchy Criterion | Real Analysis

Подписаться 139 тыс.

Просмотров 62 тыс.

50% 1

What are Cauchy sequences? We introduce the Cauchy criterion for sequences and discuss its importance. A sequence is Cauchy if and only if it converges. So Cauchy sequences are another way of characterizing convergence without involving the limit. A sequence being Cauchy roughly means that its terms get arbitrarily close to each other - no limit involved!
We'll see an example of proving a sequence is Cauchy - we prove {1/n} is a Cauchy sequence using the Archimedean property.
Cauchy Sequences are Bounded: • Proof: Cauchy Sequence...
Proof Convergent Sequences are Cauchy: • Proof: Convergent Sequ...
Proof Cauchy Sequences Converge: • Proof: Cauchy Sequence...
#Math #RealAnalysis
★DONATE★
◆ Support Wrath of Math on Patreon for early access to new videos and other exclusive benefits: / wrathofmathlessons
◆ Donate on PayPal: www.paypal.me/...
Thanks to Robert Rennie and Barbara Sharrock for their generous support on Patreon!
Thanks to Crayon Angel, my favorite musician in the world, who upon my request gave me permission to use his music in my math lessons: crayonangel.ba...
Follow Wrath of Math on...
● Instagram: / wrathofmathedu
● Facebook: / wrathofmath
● Twitter: / wrathofmathedu
My Music Channel: / @emery3050

Опубликовано:

13 окт 2024

Ссылка:

Скачать:

Готовим ссылку...

Добавить в:

Мой плейлист

Посмотреть позже

Комментарии : 80

@lightnightsky 3 года назад

Just wanted to let you know you're a life saver, I was at the brink of thinking I might have to quit math until I discovered your videos and started understanding so much that went over my head from my lectures

@gvnbrn 2 года назад

Having looked at various resources to get Cauchy sequences clear in my head, this is the first one I've truly understood. Thank you so much! Your explanation is super-clear.

@WrathofMath 2 года назад

Thanks a lot, I'm so glad it helped! Check out my analysis playlist for more! ru-vid.com/group/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxl

@jbhagawati9536 2 года назад

Absolutely

@Rsingh1 2 года назад

I agree

@santi044 Год назад

Your videos are just brilliant every time. Honestly gives me so much hope every time I feel like I wanna quit Math, because it proves to myself that I actually can understand these theorems. Thank you man.

@WrathofMath Год назад

Thanks so much for the kind words! I’m continuing to work on my real analysis playlist, I hope you’ll find many more of the videos helpful!

@santi044 Год назад

@@WrathofMath I'm glad to hear that. Keep up the good work!

@jinglepinglepie611 8 месяцев назад

Dude, you are seriously a gifted teacher. Thank you for your hard work.

@hansgans7066 3 года назад

Seriously this is next level, I just listened without pause the video to think about what you said. THANK YOU VERY MUCH. Keep going on with this, you are talented!!!!!!!!!!!!!!!!!!!

@saheedadisa8077 8 месяцев назад

First time to understand this definition for the past 5 years I have been coming across it in my academic journey. Thank you!

@WrathofMath 3 года назад

Thanks for watching! Learn more in my Real Analysis playlist: ru-vid.com/group/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli Support on Patreon to help me continue producing lessons and for early access to select videos! www.patreon.com/join/wrathofmathlessons

@EvrYRAp Год назад

HOLYYYYYYYYYYY I HAVE BEEN BINGEWATCHING YOUR VIDEOS SINCE 8HOURS AGO AND YOU ARE SO GOOD AT IT, THANK YOU

@mcalkis5771 Год назад

Brilliant. Absolutely brilliant. The best explanation of Cauchy sequences I've come across

@WrathofMath Год назад

Thanks so much! Check out my real analysis playlist for more, and let me know if you have any requests! ru-vid.com/group/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli

@anthonytaylor6297 6 месяцев назад

Wow, I was just taken for a ride.. a wild ride. But thank heavens I did, that was amazing. I'm taking mathematical analysis and it's been tough for me to understand, kinda feeling like I shouldn't be there. We aren't onto this topic at the moment but on uniform continuity. You broke that down so well, that I now feel empowered because I understood all your steps. It was clear, well-structured and beautiful. Thank you so much for this wonderful piece. Take care.

@aris.konstantinidis 3 года назад

Finally, some motivation and context regarding this topic.. much appreciated! :)

@WrathofMath 3 года назад

My pleasure, thanks for watching! If you're looking for more analysis, check out my playlist! ru-vid.com/group/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli

@elrichardo1337 3 месяца назад

getting ready to take real analysis this fall, these videos look like they're gonna be a great help - thanks in advance!

@WrathofMath 3 месяца назад

I hope they will be! Good luck and let me know if you ever have any questions! I don't know what text you'll be using, but my playlist mostly follows Real Analysis by Jay Cummings and Understanding Analysis by Abbott, though the playlist is still far from complete.

@victorvelasco9446 3 месяца назад

if ur studying before you even start to take it, you're already set

@schenzur 3 года назад

Thanks for creating this video and creating the whole real analysis playlist!

@pantelisvasileiadis2858 2 года назад

thank you very much man. Amazing explanation. I finally get a good understanding of this

@MrHolmes7 11 месяцев назад

You are amazing. Please keep doing what you do. You make the language of math as easy as the ABC song. Is there any reason you used 2/ep.... or can one just choose any epsilon? I see now...Thanks again

@abcdefghijklnopqrstuvwxyz5259 Год назад

I cannot thank you enough for this...had almost given up in trying finding a video on this topic 💕💕

@WrathofMath Год назад

So glad to help! I have lots more real analysis videos in my analysis playlist, and many more in the works!

@sinabirkeland1026 2 года назад

Thank you so much this was an amazing video:D Finally I could understand the Cauchy-criteria fully. Love from Norway (Sina and Natnael Tesema)

@drewjackson7949 2 года назад

Very informative video with excellent examples.

@yogitajindal3149 Год назад

Excellent explanation. Really very helpful Thank you so much

@yogitajindal3149 Год назад

Very easily understandable explanation Thank you so much It helped too much

@WrathofMath Год назад

Awesome! Thanks for watching!

@delzeltyna Год назад

i swear unis make everything complicated when it doesn't have to be! thank you so much!

@WrathofMath Год назад

Glad to help! Check out my analysis playlist for more and let me know if you have any questions! ru-vid.com/group/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli

@egegulce5577 9 месяцев назад

i love you, im at one of the best universities in the world studying comp sci and no prof explains this like you lmao

@WrathofMath 9 месяцев назад

Thank you! Good luck in your studies, I'm happy to be able to help!

@augusthuangmx Год назад

Thank you so much for the video!!! Really helps me a lot.

@dabeeramir1407 11 месяцев назад

Oh my god, thank you so much for this you really saved me 🙏

@WrathofMath 11 месяцев назад

Glad to help!

@maxcodez 9 месяцев назад

such an amazing video

@WrathofMath 9 месяцев назад

Thank you!

@poissontutor8524 3 месяца назад

This is super duper great!❤

@davieskayombo9106 6 месяцев назад

You are just the best

@Rluo42069 3 года назад

hi there, thanks for the video. I'm just still a bit confused on why we need to use the Archemedian property and how to know when to use this property in proofs. Thanks again

@chrisdowdy5212 6 месяцев назад

your videos are amazing

@WrathofMath 6 месяцев назад

Thank you so much 😀

@SELIMofficial64 2 года назад

Outstanding

@siavashazimi3633 Год назад

04:35 the information is not true. " In every metric space, each convergent sequence is a Cauchy sequence. The converse is not true in general."

@ScottyLane_ 2 года назад

Thank you so much! New subscriber

@ThenSaidHeUntoThem 2 года назад

Thank you for this video.

@WrathofMath 2 года назад

Thanks for watching!

@mohamedbenjbara8663 Год назад

thank youuu for the approach

@WrathofMath Год назад

My pleasure. thanks for watching!

@STG642 2 года назад

Great explanation 👍👍

@WrathofMath 2 года назад

Thanks, Shadab! Let me know if you have any questions, and check out my analysis playlist for more! ru-vid.com/group/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli

@siavashazimi3633 Год назад

By the way the video was great. Thanks a lot

@kvitka_lily 11 месяцев назад

Thank you!!!!!💓💓💓

@WrathofMath 11 месяцев назад

You are so welcome!

@SephirothVG Год назад

@14:21 You say that replacing m and n with big N which is smaller would make the sum BIGGER...Dont you mean smaller? @Wrath of Math

@MUSHIN_888 2 года назад

at 6:20, why is m,n > 21 not m,n => 21?

@kelvinndereba5423 3 года назад

Thanks for the video

@WrathofMath 3 года назад

You're very welcome, thanks for watching! If you're looking for more real analysis, check out my playlist! ru-vid.com/group/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli

@jbhagawati9536 2 года назад

Thank you so much

@WrathofMath 2 года назад

Glad to help! Thanks for watching and check out my analysis playlist if you're looking for more! ru-vid.com/group/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli

@lippy21 2 года назад

thank you so much !! :)

@kadagaladileep1862 2 года назад

help ful🌼🦋

@kadagaladileep1862 2 года назад

Thanks😁

@WrathofMath 2 года назад

You're welcome, thanks for watching! If you're looking for more analysis, check out my playlist, and let me know if you have any requests! ru-vid.com/group/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli

@KurdaHussein 2 года назад

every converge is cauchy but not conversely. ex: 1/n on R-{0}

@krishnabahetra4641 2 месяца назад

12:32, this is not correct to say that after applying the triangle inequality, it is not necessary for it to be less than epsilon.

@WrathofMath 2 месяца назад

I do not understand your comment, could you rephrase?

@ochenambrose2121 3 года назад

Thank u 🙏🏾🙏🏾

@WrathofMath 3 года назад

My pleasure, thanks for watching! If you're looking for more real analysis, check out my real analysis playlist! ru-vid.com/group/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli Many more lessons to come, let me know if you ever have any requests!