Prove the Triangle Inequality (Real Number Case)

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Real Analysis Proofs, Video #7 (Fundamental Inequalities V). Use the formula for the absolute value function and basic lemmas. It is best to structure the proof of the triangle inequality through the use of lemmas.

(0:00) Statement of the triangle inequality
(1:13) Define |x|, the absolute value of a real number, with a piecewise formula
(2:42) Lemma 1: -|a| is less than or equal to a is less than or equal to |a|
(3:34) Lemma 2: Inequalities are preserved by addition
(4:13) Lemma 3: |a| is less than or equal to c if and only if a is between -c and c
(5:22) Prove the triangle inequality using the lemmas
(9:16) Prove Lemma 1 using cases
(11:15) Prove Lemma 2
(12:24) Prove Lemma 3

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  1. Bill Kinney
  2. real analysis
  3. triangle inequality
  4. triangle inequality theorem
  5. why is the triangle inequality theorem true
  6. why is the triangle inequality true
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Комментарии
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Idk how people learned math without Youtube. Thanks!

ddiq
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There is a trend among mathematics communicators to approach proofs or mathematics in general relying on the audience to fill in gaps of explanation (essentially adopting a blasé attitude about minor details). This is ok I guess for most things but when it comes to inequalities it’s best to not skip steps in explaining. For this reason I would say this video is the best.

alephnull
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Thank you Professor, God bless you always!

DrTymish
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Sir i didnt get the define thing in the starting
|5|=5 but |-5|=-5?? How?
|| is supposed to positify the no.s within it? Isn't it?

shiwangee
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Im asking for your whatsapp line professor ... If you don't mind sir

slimthanos