Proving properties about Frobenius norms

[Yobz]

It is known that The frobenius norm, |AB|_F is less than or equal to the Frobenius norm of the two matrices, that is:
|AB|_F<=|A|_F*|B|_F
I was wondering if there was an easy way to prove this...I can't seem to work it out quite right, anyone know?

[cts]

Just standard submultiplicativity, follows from Cauchy Schwarz inequality right?

Edit: submultiplicativity is also implied just from the fact that the Frobenius norm is a matrix norm.

Edit2: actually that depends on your definition of matrix norm. I've always seen it as a submultiplicative vector norm, but I'm used to Johnson's Matrix Analysis book and I think it may be given differently elsewhere.