Banks use what are described as "adhesion" contracts. They are one-sided arrangements where one party has all of the bargaining power; e.g. if I cross out a couple things and add some interliniations on your signature card you are not going to open the account.
Accordingly, courts bend over backward to make certain the person signing an adhesion contract is treated fairly. Customers should automatically get a copy of anything they sign. If they want a copy of something they signed 5 years ago, they should get that too.
Your lawyer's worst nightmare is to give a witness on the stand a document to verify its authenticity and have the witness day, "I don't know if that's the one I signed or not; I never got a copy and the bank refused to give me one when I asked for it later."
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In this world you must be oh so smart or oh so pleasant. Well, for years I was smart. I recommend pleasant.