A proof or residency letter provides proof that the resident resides at at a specified address.
If the letter is being sent as a hard copy, then it should be in the proper business format, otherwise, if the letter is being emailed, it should have an appropriate subject line and proper grammar.
Per your request, I'm sending this formal letter to you as proof of my residency. I, [Your First and Last Name], have lived at [The Name of Your Apartment Complex, if applicable], [Your Street Address, City, State, Zip Code] since [Year you first started living there.]
I have enclosed a copy of my [Utility bill/Postal Mail/Voter's Registration Card, or Other Proof] for your records.
If you have any questions, feel free to contact me by phone at [Your Phone Number], by email at [Your Email Address], or by regular postal mail at my home address.
Thank you for your time.