This will not happen because of fractional reserve banking.
Alice deposits 1 bitcoin into the bank.
Bob deposits 10 bitcoins into the bank.
> Coins in bank: 11 bitcoins
Carpenter borrows 5 bitcoins to start up a new business.
> Coins in bank: 6 bitcoins
Don't see the problem?
Alice sees her statement with 1 bitcoin.
Bob sees his statement with 10 bitcoins.
Carpenter sees his wallet with 5 bitcoins.
Kaboom. Inflation. Fractional reserve banking will *save* us from a deflationary spiral.
Lets pretend for a moment that banks (and their insurance companies) settle on 10% reserve. So The banks can loan out some of their deposits, but have to keep at least 10% of all deposits in actual bitcoins.
Given your example the bank has 11 actual bitcoins and 16 bitcoins worth of deposits.
So the bank loans out another 9 bitcoins to Dave.
Now we have 11 coins in bank and 25 bitcoins worth of deposits.
Then the bank loans out 6 coins to Edward.
Now 11 coins in bank and 31 worth of deposits.
This process can continue until there are a total of 110 bitcoins worth of deposits.
So magically, 11 "real" bitcoins becomes 110 bitcoins worth of spending power in the economy.
Transactions between people or businesses that use the same bank don't even go through the blockchain. The bank just adjusts the necessary deposit account info accordingly.
Transactions between different banks can be acknowledged by both banks without transferring any actual bitcoins. Some people using bank A will buy from people using bank B, some using bank B will buy from people at bank A. The banks agree to keep track of net difference and on some agreed schedule (daily?) they use the blockchain to transfer only the necessary difference between them. So if in a given day bank A customers spend a total of 1,000,000 BTC in thousands of transactions with bank B customers, and bank B customers spend a total of 1,000,010 in thousands of transactions with bank A customers, there is still only a single 10 BTC transaction added to the blockchain that day to represent all that economic activity.
Why will people use banks instead of just keeping their own secure wallets? A variety of reasons. Perhaps they trust the bank to do a better job of securing the bitcoins than they believe they can do themselves? Perhaps the banks offer to pay a small interest rate on deposits? Perhaps the banks provide a convenience for engaging in bitcoin based transactions in the physical world (debit cards, checks, etc) Perhaps the banks offer reduced transaction fees (bank only has to pay blockchain transaction fees on a 10BT transaction instead of thousands of transactions valued at over 2 million BTC)?