Will a bank finance a home loan if the house appraises lower than the purchase amount?

My agent encouraged me to offer $125,000 for a house I was looking at. When I wondered if the house was worth it, she said "Don't worry. The bank won't lend you the money if the house isn't worth it.". I bought the house, and now I'm thinking she lied to me. ( It wouldn't be the only thing she was lying about.)