See, this is why I collect library cards. Library personnel are a pain in the butt sometimes, so once I can get a card somewhere I do and I never let it go. My current library card stash: Arlington County, VA; Washington, D.C.; Fort Bend County, TX; Upland, CA; Los Angeles County, CA; Cambridge, MA; Brooklyn, NY; and the Library of Congress. (I am not certain if my LOC researcher card is still valid; do those things expire?)
The Brooklyn one was the most irritating to get. They only require a piece of mail addressed to you, but I wasn't getting a lot of letters that summer and so I had to schlep back home (only ten minutes) and bring them a moving box I'd mailed to myself. Charming.
For those of you in the D.C. area, only a rent receipt is required, although it sounds like a bank statement might do. Happy reading!