Analysis - Proof of the triangle inequality by cases

This is a video that proves the triangle inequality by looking at different cases. Some steps are taken to point out the definitions that are being used so that the entire proof is strait forward and easy to follow.
Here are the different sections you jump to if you get stuck.
What cases should you consider?: 3:05
Case 1, x or y is equal to zero: 4:20
Definition of absolute value: 6:07
Case 2, x and y are positive: 7:03
Case 3, x and y are negative: 8:23
Adding numbers with different signs: 9:41
Case 4a, x is negative, y is positive, absolute value of y is larger: 11:40
Case 4b, x is negative, y is positive, absolute value of x is larger: 14:35
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[@brentonvelasquez4875]

Awesome video, awesome teacher!

[@aryaer9557]

I think there is a mistake in the second part of case4. x+y = |x|-|y| I think it is actually x+y = -(|x|-|y|) and |x+y|=|x|-|y| < x|-|y|+2|y|
then |x+y|=|x|+|y|.

[@sudhanshupandey6096]

Sir please start a calculus series from beginning. Please
Love from India.

[@fazaiacollege4020]

In case 4a, If x is negative why did you write it as positive in first step (x+y = I y I - I x I ) in this x is written as positive

[@benjaminhogan3157]

...thank you!!!...