Well : for some (such a j always exists)
Now: since the square root function is increasing and the squares are non-negative.
Finally and so on ... , thus:
and taking square roots on both sides we get the other inequality.
(1) They probably want to prove that the metrics are equivalent... by showing that the identity between them is a homeomorphism (hence they give rise to the same topology). And going through these inequalities is a simple mean of doing so.