Putting the product in the cart without reloading the page on ajax

You need to add the product to ajax with verification of its availability in the database. That is, there is a field where the user himself can enter the amount he needs before adding to the basket, then he clicks to add to the basket and if the entered amount exceeds the amount available (in the database), then issue a notification nearby (so much is not available), and also if he does not enter an arbitrary amount, but simply click to add the same product to the basket (by default 1 pcs.), to the limit (available) also give a notification.

I think I understand. An example of implementation can be found here Text hidden There is a product at the moment only 10 pieces, click on the button in the basket, the system is the same!
And yes, by adding the item to the cart would display a window that the item was added to the cart...well, like on that site.
I will send files of my project with the DB dump as we agree!