Proof of the Triangle Inequality  

 

     Notice that  and that .

 

     Adding the two statements together, we have that

 

          .

 

     By the property of absolute value, we have that

 

          , and so we are done.

 

 

Alternative Proof of the Triangle Inequality:

 

     It suffices to show that . (Afterwards, we could take the square root of both sides.)

 

           

 

 

Proof of the Reverse Triangle Inequality:  

 

     Without loss of generality, assume that |a| ≥ |b|. (If |b| ≥ |a|, then we could just swap the role of a and b in the inequality.) So it suffices to show that .

 

     However, we can write a as a  b + b, ie think of it as a = (a  b) + b.

 

     This means that . But,  by the Triangle Inequality, and so we have that . Now, subtract |b| from both sides. This gives us:

          .

 

     This is precisely what we wanted to show, and so we are done.