cash means cash. use a bank statement, IRA or brokerage statement if you can, you are not showing where the $$ is coming from, just showing you can close. it would suck to miss a good deal on a technicality like if they don't like your POf letter for whatever reason...
as for competition, the ones that are priced right are gone in under a week here, I had several I missed before I could even get in to look. when I was fast enough, ONCE, I put in no contingincies, all cash, 10% earnest, offered substantially over list and still got beat out. The guy that got it paid 7k more than my offer and probably still made 40. That makes him smart and me cheap I guess, lol.
you need speed, money, and know what your doing for those. it probably doesn't hurt to have an inside track or to be lucky either.
the ones that are priced wrong at 2x their trashed current value will sit there for months, you have to watch for the price drops and be ready to move when the market reality and the list price get relatively close. for some reason I've been able to make those work pretty well but it's a waiting game.