How to program a new key for a 2003 BMW 330i.

Hello every body .
Im in need of some help please, it all started when my original key stopped locking and unlocking the car it only starts it , because im guessing the battery is dead on it so I bought a new virgin uncut original key from ebay i took it and got it cut , then took the key to my mechanic so he can try programming it which he did out of a book, now it locks and unlocks the car but when you put it in the ignition it starts the car for a minute then shuts off by itself, is there a way i could fix my new key or did we miss a step ?

You can follow these steps:

1. Close all of the doors, the trunk and the hood.
2. Place the key in the ignition and turn it to position one, "Accessory". Turn it back to "Off". Do this four more times and remove the key from the ignition when you are done.
3. Press and hold the "unlock" button and then press and release the "lock" button three times. Release the "unlock" button. If the locks cycle, then you have done this correctly. Repeat this step for each key you wish to program.
4. Inset the key in the ignition and turn it to "Accessory" to exit programming mode.