Proof of The Triangular Inequality

The Triangular Inequality

Theorem: The triangular inequality states that if $a$ and $b$ are any two real numbers, then $|a+b|\leq{|a|+|b|}$.............(1)

Proof:: To proof the triangular inequality involves four cases and this cases are simply obtained from the order properties of the real number system as follows:

(a) If $a\geq{0}$ and $b\geq{0}$ then $a+b\geq{0}$, so $|a+b|=a+b=|a|+|b|$

(b) If $a\leq{0}$ and $b\leq{0}$ then $a+b\leq{0}$, so $|a+b|=-a+(-b)=|a|+|b|$

(c) If $a\geq{0}$ and $b\leq{0}$ then $a+b=|a|-|b|$

(d) If $a\leq{0}$ and $b\geq{0}$ then $a+b=-|a|+|b|$ then equation (1) holds in either cases.

$|a+b|=|a|-|b|$ if $|a|\geq{|b|}$
and

$|b|-|a|$ if $|b|\geq{|a|}$

QED.

In the upcoming lectures i will proof more different cases of the triangular inequality.

Thank you 

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