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For real numbers a,b: [tex]\left|a+b[/tex]|<= |a|+|b|. Is this something proven? Or is it an axiom or something?

- Thread starter JennyInTheSky
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- #1

- 8

- 0

For real numbers a,b: [tex]\left|a+b[/tex]|<= |a|+|b|. Is this something proven? Or is it an axiom or something?

- #2

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- #3

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It's a pretty important inequality. I highly suggest you convince yourself of its truth :D

- #4

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Case 1( a and b are positive):

|a+b| = a + b = |a|+|b|

Case 2 (a is positive and b is non-positive):

Let b = -y then a and y are positive. If a-y is positive:

[tex]|a+b|=|a-y| = a-y \leq a \leq |a| + |b|[/tex]

If a-y is non-positive, then y-a is positive and:

[tex]|a+b|=|a-y| = y-a \leq y = |b| \leq |a| + |b|[/tex]

Case 3 (a and b are negative):

Let a = -x, b = -y:

|a+b| = |-(x+y)| = x+y = |a|+|b|

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