You're not accounting for fractional reserve banking, where the banks only need to keep a certain amount of deposits on hand, and can loan the rest out to other people. For example, you deposit $100 into a bank. They keep, say, $20, and loan Erin $80 out to another person. This essentially creates money out of thin air since you still legally own that $100 dollars, and but now Erin has $80, bringing the total amount of money in our little economy to $180.

This all works fine assuming you don't try to withdraw all $100 since they only have $20 on hand. Nor would it be good if Erin defaults on her debt. Or the bank collapses. This is what FDIC is there to protect, where the government will create new money in case things collapse.