Just my take on it.

Let's look at the structure of an e-mail or a letter for that matter.  Simply said you have 3 parts.

1. opening or greeting

2. body

3. closing or ending

NOTE:  greeting means hello and not good-bye.

As was pointed out " yours faithfully " is out of style but it still can be found in use especially in British English. However, as was also pointed out, there are alternatives that are used in both British and American (North American) English.

1. More formal ---   Sincerely yours, Yours sincerely, or even just Sincerely

2. less formal--- Sincerely, Kind/Best/Warm regards, Regards, *Best wishes

3. informal --- Best (wishes), Regards, * Just write your name *, See you, Thanks, etc., etc.

Not that this is a rule but in American English a comma is generally used after the greeting and after the closing.