Proof: A Useful Absolute Value Inequality | Real Analysis

For real numbers a and b, |a| is less than or equal to b if and only if -b is less than or equal to a is less than or equal to b. We'll prove this useful theorem about the absolute value in today's real analysis lesson!
This theorem gives us a way to go from an absolute value inequality (|a| is less than or equal to b) to a nice inequality with no absolute value! We can of course also use the other direction of the theorem in order to PROVE an absolute value relationship. If we want to prove that |a| is less than or equal to b, we just need to show that -b is less than or equal to a is less than or equal to b. That is the power of this result, it helps us work with absolute value! The basic idea is that if |a| is less than or equal to b, it must be the case that -b is more negative than a and b is more positive than a. Thus, whether a is positive or negative, its magnitude is less than b. Notice if b is greater than or equal to |a|, then b must be non-negative.
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[@WrathofMath]

Support the production of this course by joining Wrath of Math to access exclusive and early videos, original music, plus the real analysis lecture notes at the premium tier! kzbin.info/door/yEKvaxi8mt9FMc62MHcliwjoin Real Analysis course: kzbin.info/aero/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli Real Analysis exercises: kzbin.info/aero/PLztBpqftvzxXAN05Gm3iNmpz9SkVfLNqC

[@alexadinolfi6038]

I'm taking real analysis this semester and I think this channel is going to be super helpful. Gotta say this was probably the most straightforward explanation of the proof of this theorem I've seen, it's very easy to understand. I have a feeling I'll be spending a lot of time watching your videos this semester. Keep up the good work!

[@thedadbrains]

Thank you so much, these videos are literally life saving.

[@Justamemekid959]

Brilliant... Thanks for this... Keep it up Sir!

[@WrathofMath]

Thank you, I will! Thanks for watching and let me know if you ever have any video requests. If you're looking for more analysis, check out my playlist! kzbin.info/aero/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli

[@baijokull]

Think I might be watching at least 76 more of your videos if I'm going to pass this class :D

[@WrathofMath]

Good luck! Let me know if you have any questions, there is still a lot I need to add to this Real Analysis playlist but it will definitely help you for a bit!

[@ProofCare]

Finally found a good video explaining absolute value inequality!🙏 From India

[@WrathofMath]

Glad it helped! Thanks for watching!

[@killuaenthusiast]

Thank you! This is so helpful.

[@WrathofMath]

Glad to hear it! Lots more real analysis lessons coming, let me know if you ever have any video requests!

[@martinperu6207]

Thank.. What book of Mathematics for basic in proof and real analysis?

[@existentialrap521]

This is a certified hood CLASSIC. Gonna be bumpin this proof a lot

[@bobmarley9905]

bruh 😂

[@existentialrap521]

@@bobmarley9905 We survived. Barely. Advanced Calc I beat my ass. Good stuff tho. I liked it. Challenged me af. First real proof course.

[@minalidharmasena5764]

you are explaining it so clearly sir...thank you so much

[@WrathofMath]

So glad it helped, thanks for watching! If you're looking for more analysis, check out my playlist! kzbin.info/aero/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli

[@bethelyemane6578]

can some one explain it easily am not getting it,??

[@hz6612]

Thank you so much !!! you gave a really great explanation !!

[@WrathofMath]

Glad it was helpful!

[@ARABELAFRATEA]

Hello,You are doing a great job! Can we hope for some videos explaining measure theory and complex analysis? Thanks a lot!

[@WrathofMath]

Thank you! Yes, but it will be some time before I get to that. Current focuses are Real Analysis, Abstract Algebra, and Graph Theory. And I teach two courses of Calculus, so I do that a lot just because.

[@kelvinthehuman]

Amazing. Hope we all find a way to pay you well, buddie. for what you're doing. This is a great contribution to education, it helps keeping a nice academic atmosphere. Thanks a lot.@@WrathofMath

[@AyandaMakhetha]

I'm just studying for algebra😭 non too crazy yet, thanks for this!

[@WrathofMath]

Happy to help!

[@tuptge]

Thanks, excelent video and presentation!

[@janosmarton7895]

You should consider ASMR discrete maths explanation videos ":D your jokes are killing me :D

[@user-dp9yn7zf4l]

The video helps me realize I been using it without knowing the proof

[@mouradbouderbala4709]

Greetings from Algeria, thanks for the effort. I hope your creativity continues

[@HafeniShitaku]

Other examples

[@nobirapals974]

*sigh* I use the second direction to prove first direction and got confused on what to do next lol. silly me. thank you for the cool vid!

[@nobirapals974]

oh, and I just want to make sure -|a|

[@WrathofMath]

Thanks for watching! -|a|

[@nobirapals974]

@@WrathofMath ah i see, thank you so much! you earn yourself a new subscriber :)

[@mat-on-go8644]

i have a video request for this question 4|x| < 7x − 6. thank you

[@vickimazur8]

| x - r | < 1/n with r in reals, n in naturals and r in rationals?

[@danielrocha2521]

thank you! how can we prove this by contradiction doe

[@Sukmato]

The ending music on every single video is lit🔥

[@maxpercer7119]

i see a chessboard, do you teach chess theory?

[@dynamix9916]

2:02 Finally, it gets into the proof.

[@ezenaboifeyinwa]

Please solve 🙏 |x| + |x+1| + |1-x|

[@RealEverythingComputers]

Great video thanks!

[@WrathofMath]

Thanks for watching!

[@سارهداودسلمانأحمد]

Hello sir, how can I prove that |a|

[@WrathofMath]

Thanks for watching, and you cannot prove that - consider any negative value of a for a counterexample.

[@monikanarayan5820]

Your videos are very helpful. Can you please tell me the textbook that you refer to?

[@WrathofMath]

I'm glad they've been helpful, thanks for watching! I don't know if I refer to a specific textbook in this video, but I am crafting my Real Analysis playlist primarily following three textbooks. Be sure to check out the playlist if you haven't many more lesson to come: kzbin.info/aero/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli Here are the textbooks, in order of importance to me: Real Analysis by Jay Cummings Understanding Analysis by Stephen Abbott A Friendly Introduction to Analysis by Kosmala

[@isurudilshan7172]

thank u very much

[@WrathofMath]

My pleasure, thanks for watching!

[@mat-on-go8644]

hi thank you for the solution

[@mahmoudalbahar1641]

Many thanks for this good video.

[@benjaminhogan3157]

Is this proof the same for a strict less than inequality? Thx

[@WrathofMath]

Yes indeed!

[@cheatyhotbeef2636]

I just stumbed upon this video, so I'm not doing real analysis

[@jrtthamasha7306]

Thankyou Sir🥰

[@WrathofMath]

Glad to help!

[@krishkaru1100]

Thank you so much❤

[@WrathofMath]

Glad to help - thanks for watching!

[@hanaalhomrani7040]

Thank you it was helpful

[@WrathofMath]

Glad to hear it, thanks for watching!

[@thatpilotguy4995]

Geez, stop bouncing all over the place. Just stand still. Gave me a headache

[@documentemail7299]

Why we can't do case work in first part of proof? Can't we start with-|a|

[@jeremiahv6498]

Why is this question not answered? It's been a year.

[@agustincai]

@@jeremiahv6498 did you get the answer?