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.