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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Пікірлер: 59
Check out my Real Analysis playlist to see just how useful this result is! kzread.info/head/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli
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!
Thank you so much, these videos are literally life saving.
Thanks, excelent video and presentation!
Many thanks for this good video.
Greetings from Algeria, thanks for the effort. I hope your creativity continues
Finally found a good video explaining absolute value inequality!🙏 From India
@WrathofMath
3 жыл бұрын
Glad it helped! Thanks for watching!
you are explaining it so clearly sir...thank you so much
@WrathofMath
3 жыл бұрын
So glad it helped, thanks for watching! If you're looking for more analysis, check out my playlist! kzread.info/head/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli
Thank you! This is so helpful.
@WrathofMath
3 жыл бұрын
Glad to hear it! Lots more real analysis lessons coming, let me know if you ever have any video requests!
This is a certified hood CLASSIC. Gonna be bumpin this proof a lot
@bobmarley9905
Ай бұрын
bruh 😂
@existentialrap521
Ай бұрын
@@bobmarley9905 We survived. Barely. Advanced Calc I beat my ass. Good stuff tho. I liked it. Challenged me af. First real proof course.
Brilliant... Thanks for this... Keep it up Sir!
@WrathofMath
3 жыл бұрын
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! kzread.info/head/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli
Thank you so much !!! you gave a really great explanation !!
@WrathofMath
7 ай бұрын
Glad it was helpful!
Thank.. What book of Mathematics for basic in proof and real analysis?
thank you! how can we prove this by contradiction doe
The ending music on every single video is lit🔥
i have a video request for this question 4|x| < 7x − 6. thank you
The video helps me realize I been using it without knowing the proof
Hello,You are doing a great job! Can we hope for some videos explaining measure theory and complex analysis? Thanks a lot!
@WrathofMath
Жыл бұрын
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
9 ай бұрын
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
hi thank you for the solution
Thank you it was helpful
@WrathofMath
3 жыл бұрын
Glad to hear it, thanks for watching!
You should consider ASMR discrete maths explanation videos ":D your jokes are killing me :D
can some one explain it easily am not getting it,??
Your videos are very helpful. Can you please tell me the textbook that you refer to?
@WrathofMath
3 жыл бұрын
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: kzread.info/head/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
Thank you so much❤
@WrathofMath
Жыл бұрын
Glad to help - thanks for watching!
thank u very much
@WrathofMath
3 жыл бұрын
My pleasure, thanks for watching!
| x - r | < 1/n with r in reals, n in naturals and r in rationals?
*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
2 жыл бұрын
oh, and I just want to make sure -|a|
@WrathofMath
2 жыл бұрын
Thanks for watching! -|a|
@nobirapals974
2 жыл бұрын
@@WrathofMath ah i see, thank you so much! you earn yourself a new subscriber :)
Thankyou Sir🥰
@WrathofMath
Жыл бұрын
Glad to help!
I'm just studying for algebra😭 non too crazy yet, thanks for this!
@WrathofMath
3 ай бұрын
Happy to help!
Is this proof the same for a strict less than inequality? Thx
@WrathofMath
2 жыл бұрын
Yes indeed!
Hello sir, how can I prove that |a|
@WrathofMath
2 жыл бұрын
Thanks for watching, and you cannot prove that - consider any negative value of a for a counterexample.
Why we can't do case work in first part of proof? Can't we start with-|a|
@jeremiahv6498
Жыл бұрын
Why is this question not answered? It's been a year.
@agustincai
Жыл бұрын
@@jeremiahv6498 did you get the answer?
Other examples
Please solve 🙏 |x| + |x+1| + |1-x|
i see a chessboard, do you teach chess theory?
I just stumbed upon this video, so I'm not doing real analysis
Geez, stop bouncing all over the place. Just stand still. Gave me a headache