How to use the phrase the Internet correctly

Correct phrase	Common error
I'm on the Internet.	I'm in the Internet.
I'm on the Net.
I surf the Internet.	I surf in the Internet.
I surf the Net.
You'll find the information on the Internet.	You'll find the information in the Internet.
In our hotel all rooms have access to the Internet.	In our hotel all rooms have access to Internet.
I use the Internet.	I use internet.
You can buy the book over the Internet.	You can buy the book in the Internet.
I'm online.

The word Internet is often written in capital letters.