Answer
See the proof below.
Work Step by Step
Since $-|a| \le a \le |a|$ and $-|b| \le b \le |b|$, then by addition we have
$-|a|+-|b|\le a+b\le |a|+|b|$
$-(|a|+|b|)\le a+b\le |a|+|b|$
This can be rewritten as:
$|a+b|\le |a|+|b|$
which is the triangle inequality.
