How does fractional banking allow there to be an increase in the money supply?

Banks can lend up to 90% of money on deposit in the bank to other clients.
The other 10% that the bank does not lend is called the "fractional reserve".

Now, by lending this 90% to a client, the money supply may increase based on the procedure below:
First client deposits 100$ in the bank
Bank lends 90$ to second client who deposits these 90$ in another bank
This other bank lends 81$ from the deposited 90$ to a third client who deposits them in a new bank.
The new bank lends 72.9$ out of these 81$ to a fourth clients who deposits them in a different bankand so the cycle continues increasing the money supply