So Alice doesn't send Bob w. Bob sends back a challenge c, saying prove it. In doing the check, Bob uses Alice's public key h. If Alice doesn't know the secret key x corresponding to h, then the verification will fail. We are ready to do digital cash.
Digital cash is a signed payment message.
This fact tells us right away two problems with digital cash. The first is that, signed or not, the cash is a computer string, a series of 0s and 1s. It's easy to make a copy of it. It's easy to make a million copies of it. It's analogous to counterfeiting.