That's not the triangle inequality that they proved. The proof is fine.
the question and prove:
http://i33.tinypic.com/2gvk7k9.jpg
they use the triangle innequality to prove itself
??
Problem: Prove that
Proof:
Lemma:
Proof: Since the above is equivanlent to . Expanding both sides gives . Noticing that we may rewrite this inequality as or which is trivially true.
Particularly the above tells us that or equivalently .
Remark: Since we could have equally as well proven that we may actually strengthen this claim to .