Intro to Cauchy Sequences and Cauchy Criterion | Real Analysis

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
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[@lightnightsky]

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[@gvnbrn]

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]

Thanks a lot, I'm so glad it helped! Check out my analysis playlist for more! th-cam.com/play/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxl.html

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[@WrathofMath]

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!

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[@saheedadisa8077]

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

[@WrathofMath]

Thanks for watching! Learn more in my Real Analysis playlist: th-cam.com/play/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli.html
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[@mcalkis5771]

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

[@WrathofMath]

Thanks so much! Check out my real analysis playlist for more, and let me know if you have any requests! th-cam.com/play/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli.html

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[@aris.konstantinidis]

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

[@WrathofMath]

My pleasure, thanks for watching! If you're looking for more analysis, check out my playlist!
th-cam.com/play/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli.html

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

[@WrathofMath]

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.

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

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

[@MrHolmes7]

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

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[@WrathofMath]

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

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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]

Very informative video with excellent examples.

[@yogitajindal3149]

Excellent explanation. Really very helpful Thank you so much

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[@WrathofMath]

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[@WrathofMath]

Glad to help! Check out my analysis playlist for more and let me know if you have any questions! th-cam.com/play/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli.html

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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]

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[@WrathofMath]

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[@WrathofMath]

Thank you!

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[@Rluo42069]

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

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[@WrathofMath]

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[@siavashazimi3633]

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."

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[@ThenSaidHeUntoThem]

Thank you for this video.

[@WrathofMath]

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[@mohamedbenjbara8663]

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[@WrathofMath]

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[@WrathofMath]

Thanks, Shadab! Let me know if you have any questions, and check out my analysis playlist for more! th-cam.com/play/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli.html

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[@WrathofMath]

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[@SephirothVG]

@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]

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

[@kelvinndereba5423]

Thanks for the video

[@WrathofMath]

You're very welcome, thanks for watching! If you're looking for more real analysis, check out my playlist! th-cam.com/play/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli.html

[@jbhagawati9536]

Thank you so much

[@WrathofMath]

Glad to help! Thanks for watching and check out my analysis playlist if you're looking for more! th-cam.com/play/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli.html

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Thanks😁

[@WrathofMath]

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! th-cam.com/play/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli.html

[@KurdaHussein]

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

[@krishnabahetra4641]

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]

I do not understand your comment, could you rephrase?

[@ochenambrose2121]

Thank u 🙏🏾🙏🏾

[@WrathofMath]

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

[@hammadmhadi8135]

♥️

[@WrathofMath]

Thanks for watching!

[@diliniuthpala1802]

How getN>2/£

[@Yamiku]

It is explained at 12:06

[@egegulce5577]

fucking legend

[@winstonrodney9929]

Thank you for the video but you didn't explain how you got your 1/10

[@drewjackson7949]

It was just chosen arbitrarily to show that the Cauchy inequality holds.