Take another look at the passage:
Ralph spent all of his savings on a diamond ring to give to his girlfriend Melanie. A greedy young woman who had a box full of expensive trinkets from ex-boyfriends. Melanie was always looking for the man with the fattest wallet.
You chose the third part, Melanie was always looking for the man with the fattest wallet. This part has a main clause: Melanie is the subject; was looking is the verb. A fragment will not have a main clause.