How to Prove a Sequence is a Cauchy Sequence Advanced Calculus Proof with {n^2/(n^2 + 1)}

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How to Prove a Sequence is a Cauchy Sequence Advanced Calculus Proof with {n^2/(n^2 + 1)}

[@amaliacoughlan7071]

I don't know if you're aware of the incredible impact you have on students by uploading videos to CZcams, but I'm sincerely grateful. My professor is a math prodigy, but he cannot teach. You've been the only source of help I've gotten for my real analysis class.

[@TheMathSorcerer]

Thank you!!

[@imonsanyal]

Once a while, you come across an explanation so perfect, it makes you realise how easy the topic that you never really understood actually was.
This is that video!

[@kumailkhaki266]

The word everybody is looking for is Intuition! You've shown your intuition, which helps us see WHY you're doing certain things, it clarifies your steps in achieving the desired goal and that's something that teachers fail to do. They teach as if we're inside their brains! Thank you!

[@TheMathSorcerer]

You are welcome!

[@gisellabradley7270]

You are very clear in all your steps. It makes it easy to understand! Thank you so much!!💜

[@TheMathSorcerer]

no problem:)

[@adriana_irl]

this video was 100 times better than my professors lectures. You were able to explain this process in 10 minutes, whereas he has explained it for hours across multiple days and I never got it until now!!

[@sarahrickel8488]

This was a very well done and helpful video! Thank you for showing everything so thoroughly!

[@TheMathSorcerer]

thank you! so happy it helped someone!!

[@InfinityMW2]

Hey I finally understood Cauchy sequences after a few days. Im taking Real Analysis right now and its been tough but rewarding so far

[@vdav17]

Thanks for this, one question, is it valid to go further in the inequalities and write that 1/m^2 + 1/n^2

[@TheMathSorcerer]

Yes, that is much better, much cleaner! I like your way:)

[@tonyreyes8190]

Thank you! Very helpful!. I’m going to watch more and if you make more I’ll keep on watching these. You make it so easy for me to understand! I love that feeling when things click and you just learned something new. That is a gift and I appreciate so much. Thank you!

[@TheMathSorcerer]

Yes I love that feeling too! Yes soon I will be posting more proofs❤️

[@Karim-nq1be]

Really nicely explained, I barely use english in my daily life, but I must say that was one of the most helpful explanations I've seen on how to prove a sequence is Cauchy.

[@jonathanmolina7690]

Great video! You have no idea how much this helps!

[@TheMathSorcerer]

awesome!

[@d00b49]

Thanks for the great video! I follow it all except for one step. How do you know e/2 is a suitable choice for each term? The requirement of |a_n - a_m| < e can be still be satisfied if one term is larger than e/2 as long as the other is less than the difference between 1 and the first term.

[@fabioutrilla7109]

Hey, I'm no expert but I was watching an earlier video of his and the reason he picks e/2 is because you want the whole sum of 1/m^2 + 1/n^2 to be less than epsilon so if you can prove that each individual part(1/m^2 by itself) is less than e/2 then you satisfy this.

[@Sipnayan]

This is really a clear explanation. New subscriber here. Keep it up.

[@TheMathSorcerer]

Thanks 😄

[@frankie8086]

Thank you so much for this!!! I’m a college student that was taking real analysis, complex analysis, and abstract algebra, with a writing intensive course. All those classes combined, was way too much for me. Especially since I work. The class I dropped was real analysis bc it was eating up most my time. I’m sad I wasn’t able to finish it this semester but hopefully next semester I’ll be able to take it with less of a workload. I’ll def be looking through your vids in preparation! Flawless teaching!

[@TheMathSorcerer]

:)

[@mptapasdas]

BEST EXPLANATION. THANKS SIR

[@TheMathSorcerer]

thank you!

[@sourceOfJoy21]

Thank you so much this was helpful

[@Fortune_Crown]

WOWWWW!!!! THIS IS BEAUTIFUL! Thank you! ❤❤❤

[@taronmamyan4972]

Thank you, CZcams man, for your contribution to society.

[@thembanovela4273]

Thank you so much. I am a big fan of your work... salute!!!

[@TheMathSorcerer]

You are welcome!

[@sonic5d]

This was beautifully done. Thank you. I have subscribed :D

[@TheMathSorcerer]

Awesome, thank you!

[@ЕвгенияГанжа]

Thanks a million! I have subscribed to your great channel! :)

[@zacharychild2993]

Thanks for the video! Got a midterm tomorrow

[@TheMathSorcerer]

good luck!

[@pablolemus2983]

could you go over how you went from n^2> 2/ε -> 1/n^2 < ε/2 because I don't understand how your dividing form N^2 and 2/ε?

[@elijah6218]

Amazing video, thanks for this great content!!

[@TheMathSorcerer]

you are welcome!

[@peterschannel2055]

Thank you very much. Good video.

[@TheMathSorcerer]

you are welcome:)

[@dmitarzvonimirmitev6644]

Very nice video ! Thank you!

[@TheMathSorcerer]

you are very welcome!!

[@burakergocmen5661]

thanks , really straightforward

[@TheMathSorcerer]

glad it helped:)

[@Modi_M4]

Thank you so much 🥺♥️♥️♥️♥️

[@TheMathSorcerer]

glad it helped!!!

[@graceododo9481]

Very impactful.....thank you so much sir

[@diyagoyal6685]

Really Helpful in my math assignment. :-)

[@vijaysinghchauhan7079]

Just beautiful

[@lemyul]

thanks

[@kaizoku4life]

Thank you very much

[@TheMathSorcerer]

👍

[@mijajchokker1459]

Hey, thanks 💜

[@elizabethnziokikatunge2011]

thanks dude

[@annixu9219]

Nice vedio, but how about 1/ sqrt(n(n+1)) which is diverges?

[@asmakhalid2356]

excellent. Just that auto write on the bottom of the video bothers me a lot. is there anything that can be done about it

[@TheMathSorcerer]

thank you:)

[@tjk4261]

yes, this was indeed helpful :)

[@peterschannel2055]

Why does this not work when I apply the triangle inequality early? | n^2/(n^+1) + -m^2/(m^2+1) | < or = | n^2/(n^+1) | + | -m^2/(m^2+1) |
Then I would show that n^2/(n^2+1) is smaller than Epsilon/2 (and the right side respectively), however It seems impossible to show it this way.
Is it just down to luck to get the correct reshaping done before applying it?

[@TheMathSorcerer]

yeah you have to force it to work, you can always do things that won't work, the trick is, to make it work, that is the challenge!

[@BlaqueT]

Thanks for the video. I'm not sure if I'm missing something, but why does N suddenly become larger than sqrt(eps/2) at the end when it was previously defined as N>sqrt(2/eps) at the start of the final part of the proof?

[@TheMathSorcerer]

I don't see it sorry! I looked I can't find what you are referencing:)

[@BlaqueT]

@@TheMathSorcerer at the start of the proof (following the scratchwork), you defined N>sqrt(2/eps), but then at the end, it became N>sqrt(eps/2)?

[@TheMathSorcerer]

@@BlaqueT I still don't see it sorry lol, where at the end?

[@BlaqueT]

@@TheMathSorcerer I'll just grab the timestamp :)

[@TheMathSorcerer]

@@BlaqueT I see it,small mistake thank you!!!

[@greenrain55]

thank u king

[@TheMathSorcerer]

you are welcome:)

[@slickmello7324]

Nice video

[@TheMathSorcerer]

Thank you

[@scp_in_iitb]

Nice!!

[@nopewtfisthis]

Good stuff

[@ayanceferzade8882]

very easy thanks/

[@rishesmishra1740]

Very nice video but you made a mistake on 11:01....... Well np ,still a wonderful video

[@matthewforster4840]

the videos on 4 minutes long

[@zyrex1091]

🔥💋

[@sarahmedouni8844]

so im the only one that didn't get it, nice!

[@jingyilyu9147]

why did not my professor show step by step

[@abkonmalonma1028]

This is fucking algebra

[@dinglerdangler]

thanks