Where does the money required for reserves and loans come from? I understand that banks must keep a certain percentage of money in its reserves and it can loan the rest out. What I don't understand is how the money supply increases

For example, let's take a scenario where somebody deposits 100 dollars into a bank. So the bank has 10% of required reserves, so 10 dollars is subtracted from this 100 dollars. So 90 dollars can be used for loans. 

But wouldn't this mean that if the person who payed the 100 dollars wanted the 100 dollars back in cash, the bank has spent all the money?

Edit: I watched a video stating that the 90 dollars is created by the bank. Is this true?
Well I think that the key to understanding this is to realize that money supply as other macroeconomic aggregates must be defined over period of time. 

If you deposit 100\$ bill at a time $t$ with  10% reserve system money supply can increase up to 1000\$ because first bank keeps 10\$ and lend outs 90\$ next keeps 9\$ and lends out 81\$ etc. and sum of this infinite sequence with 10% reserves is 1000.

You are completely correct to say that if people would start withdrawing money let’s say at time $t+1$ money supply would contract. So in this example at $t-1$ money supply would be 100\$, at time $t$ assuming no borrowing constraints and perfect markets money supply would be 1000\$ (because of the multiplier) and at time $t+1$ when you withdraw the money (so there is run on a banks with this one depositor) then money supply would contract back to its original 100\$.

People keep having this misconception about money supply, that in only increases if there is some permanent irreversible increase in quantity of money but that not correct. For example, consider another macroeconomic aggregate - unemployment, you would probably say unemployment increased if a person lost job even if in a year or so that same person gets their job back. The same applies to all aggregates.