Send a letter telling client that effective mm/dd/yyyy the bank will no longer be able to service their account. They should cease making deposits allowing all items to clear. They can close account or on mm/dd/yyyy the bank will send them a cashier's check for collected balance in the account.
As previously mentioned, adjust timeline if pension direct deposits are involved.
I think there may be sample letters in Bankers Tools.
Has the bank ever been threatened by client? Oh yes. In my 30 years, I have been told more than once that they are getting a lawyer and will sue the bank. Have they ever filed suit? No. There is no basis for a suit, disclosure clearly says bank can close at any time.
One personal observation: the dirtier the client, the easier the exit. They generally do not want to make a scene (other than the occasional spouting off that the bank is run by idiots.)
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Life without Jesus is like an unsharpened pencil - it has no point.