triangle inequality

stuck on last part :nosebleed:

Q:

Use triangle inequality to prove

|x+y| \leq |x|+|y|

to prove

||a|-|b|| \leq |a-b|

A:

x=a-b
y=b

|(a-b)+b|

\leq

|a-b|+|b|

|a|

\leq

|a-b|+|b|

|a|-|b|

\leq

|a-b|

how do i get the extra modulos sign on the LHS of the last part to get the result?

(edited 5 years ago)

Original post by KloppOClock

stuck on last part :nosebleed:

Q:

Use triangle inequality to prove

|x+y| \leq |x|+|y|

to prove

||a|-|b|| \leq |a-b|

A:

x=a-b
y=b

|(a-b)+b|

\leq

|a-b|+|b|

|a|

\leq

|a-b|+|b|

|a|-|b|

\leq

|a-b|

how do i get the extra modulos sign on the LHS of the last part to get the result?

Well,
Using same argument you can show: |b|-|a|<= |b-a| = |a-b| and your result follows by combining these two results as |a| = max |a,-a|

Original post by Quantum Horizon

Well,
Using same argument you can show: |b|-|a|<= |b-a| = |a-b| and your result follows by combining these two results as |a| = max |a,-a|

boss

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