Proof: Triangle Inequality Theorem | Real Analysis

The absolute value of a sum is less than or equal to the sum of the absolute values for any two real numbers. That is: |a+b| is less than or equal to |a|+|b|. This is called the triangle inequality. It's very useful in real analysis and we'll prove it in today's lesson! The name of the theorem is explained a little below.
If we considered a triangle made of vectors x, y, and z, with x and z starting at the origin, then the vector z is the vector x+y, and thus the theorem says this third side (z) of the triangle must be no greater than the sum of the other two sides: |z| is less than or equal to |x| + |y| and since the vector z is the vector x+y we have |x+y| is less than or equal to |x| + |y|. Speaking strictly in terms of geometry, we'd often use a strictly less than inequality for this theorem because the equality case describes a "degenerate" triangle, where the "triangle" is just a line. However, the "less than or equal to" inequality is perfectly valid and reasonable for real numbers.
◆ Donate on PayPal: www.paypal.me/...
◆ Support Wrath of Math on Patreon: / wrathofmathlessons
I hope you find this video helpful, and be sure to ask any questions down in the comments!
********************************************************************
The outro music is by a favorite musician of mine named Vallow, who, upon my request, kindly gave me permission to use his music in my outros. I usually put my own music in the outros, but I love Vallow's music, and wanted to share it with those of you watching. Please check out all of his wonderful work.
Vallow Bandcamp: vallow.bandcam...
Vallow Spotify: open.spotify.c...
Vallow SoundCloud: / benwatts-3
********************************************************************
+WRATH OF MATH+
Follow Wrath of Math on...
● Instagram: / wrathofmathedu
● Facebook: / wrathofmath
● Twitter: / wrathofmathedu
My Music Channel: / seanemusic

[@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

[@thedeathofbirth0763]

This is how a person who truly understands logic, can beautifully and precisely communicate it . Thank you.

[@WrathofMath]

Thank you!

[@punditgi]

Very clear proof. Well done!

[@WrathofMath]

Thank you, glad it was clear!

[@ryantrynda3029]

why is it called triangle inequality thou

[@edisonnogalesantezana4761]

The triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side.

[@red3987]

Beautiful and concise!! thank you for helping me understand this proof better!

[@WrathofMath]

Thanks so much, glad it helped! Thanks for watching and let me know if you ever have any questions!

[@red3987]

@@WrathofMath WIll do!!!

[@androidandroid9591]

An excellent AND simple way to grasp a crucial theorem for the understanding of analysis. Txs.

[@WrathofMath]

Glad to help!

[@Alan-sv6ym]

We really love you here in south Africa 🇿🇦😭💜💜💜

[@WrathofMath]

Thank you so much! Love back from the USA's east coast!

[@benjtheo414]

if 2:44 wasn't directly obvious to you as it was for me you can use this in the proof: a1

[@punditgi]

Can you do an entire video on absolute value identities as building blocks for other theorems? I would also illustrate these both on a 1D number line and on a 2D plane as vectors.

[@davaariantara3704]

like |xy|=|x|*|y|? i think you will find it obvious if you proof it by case-by-case like above in the above video or you can even proof it by |x|=squareroot(x^2) ^ just trying to give you an idea that its not hard

[@johnsally8320]

My man, you're awesome! Much love from Toronto, Canada.

[@WrathofMath]

Thanks so much!

[@mahmoudalbahar1641]

Many thanks for this good video.

[@harjassingh7758]

thank you bro / sir appreciated !!

[@H3XED_OwO]

beautiful and simple proof!

[@WrathofMath]

Thanks for watching!

[@shhaque9448]

This guy really did a 2 hour lecture in 5 minutes

[@Bhaskar3D]

It's very good sir. I am from Assam, Indian.

[@valeriereid2337]

Thanks for explaining this clearly. If you are able to, please do some work from Abbott's "Understanding Analysis." It would be awesome.

[@husseinsslehi4428]

Very clear explanation thank you

[@WrathofMath]

Thanks for watching! If you're looking for more analysis, check out my playlist: kzbin.info/aero/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli

[@jamesjones6359]

Great explanation. Thank you.

[@WrathofMath]

My pleasure, thanks for watching and check out my Real Analysis playlist if you're looking for more! kzbin.info/aero/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli Many new real analysis lessons are coming this year, let me know if you ever have any requests!

[@TahaCodes]

Very nice and neat. Thank you. Got yourself a new sub!!!

[@ProgressiveVoice]

Thanks!

[@putin_navsegda6487]

great proof, very understandable! Thank you!

[@WrathofMath]

Thank you, glad it helped! Check out my analysis playlist for more! kzbin.info/aero/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli

[@Divine_wsk21]

Can you please do the same proof with a,b and c

[@silvo9460]

wow thank you so much sir!!!!!

[@yibanathimnyanda725]

Thank this really helped.

[@WrathofMath]

So glad to hear it, check out my Real Analysis playlist if you're looking for more, many more analysis lessons still to come! kzbin.info/aero/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli

[@samuelbassey6806]

Thanks for sharing

[@bekhaddaderrar2111]

Thank you so much for this video

[@WrathofMath]

My pleasure, thanks for watching! If you're looking for more real analysis, check out my playlist! kzbin.info/aero/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli

[@jeswintitusjv5927]

Sir , could I know why this is called as "triangular inequality" ? Is this related to triangle when viewed geometrically ?

[@WrathofMath]

Thanks for watching and the question! Yes, particularly with vectors involved. Suppose we have points A, B, and C, and then three vectors AB, CB, and AC, that make a triangle. The vector AB is the same as AC + CB if you know vector addition. And this side of the triangle, AB = AC + CB has length |AB| = |AC + CB|, and this length must be no greater than the sum of the other two sides, that is |AC| + |CB|.

[@Umar-lj5jx]

Thanks a bunch

[@anasshuaibu2530]

Very good ' but plz prove it again when â and b are real numbers.

[@TVSuchty]

Can you show that over R^n and all possible dot products, where ||a||^2 = a dot a ?

[@qedmath1729]

let a = . By definition of dot product we have a dot a = x^2+y^2. Similarly for a = , ||a|| = sqrt(x^2 + y^2) or ||a||^2 = x^2 + y^2. Hence we see that the 2 are equal.

[@TVSuchty]

@@qedmath1729 I mean that u show that the theorem holds over all dot products, where by definition ||a||^2 = a dot a

[@navjyotsinghvishen3287]

Group theory from basic 🙏

[@bonniem1865]

Thank you thank you thank you

[@WrathofMath]

You are very welcome!

[@ali-hg103]

That outro thooo🔥

[@WrathofMath]

Artist goes by Crayon Angel now - love his stuff!

[@gianlucalepiscopia3123]

Beautiful

[@WrathofMath]

Thanks for watching!

[@augustus2834]

great! the only thing that bothered me a little bit is that we didnt take into account when a number is negative and the other is positive. but that case is trivial so i guess doesnt matter that much ahahaha

[@rosepierce9382]

I think it doesn't really matter because he is only taking the sum of a + b to be negative or positive.

[@이머현]

super good

[@WrathofMath]

Thank you! If you're looking for more analysis, check out my playlist! kzbin.info/aero/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli

[@muhammadasghar7153]

I Asked question but you didn't answer me

[@pakistaninstituteofmedical7185]

Yes I also ask question but didn't reply

[@ahh61]

I asked a question and he answered me so be happy. Maybe next time ask a better question.

[@muhammadasghar7153]

Shut up loser baby hahhahah🤑🤑

[@ahh61]

@@muhammadasghar7153 At least I'm not dumb like you. Your question was probably what's 2+2. You stupid. My questions are always answered. He even answered a bunch of my questions on the last live episode.

[@WrathofMath]

I'm working on it folks, I get a lot of comments and don't have much time. If you ever come by a live stream, that is time I set aside specifically to answer your questions so you can be sure I'll respond if you come by one of those! Thanks for watching and I'm trying to catch up on the comments!

[@samborn3515]

Please I stil find this difficult

[@existentialrap521]

Wrath, once i get this mfing masters degree ima invite you to a carne asada thx G

[@daniellin188]

-(a+b) does not equal (-a) + (-b) by the way

[@Zinani-z7b]

It is Darling! Look at it again.

[@keithplays1184]

1st 😊

[@WrathofMath]

Thanks for watching, Keith!

[@samuelbassey6806]

Thanks for sharing