I need to find a check that I wrote to a bank in 1988 from a bank that
has since been bought.  Both banks have been bought out, and their
buyers were bought out.  I have the bank name and account number at
the time.  How do I go about finding that check?

Even if the banks had not been bought out, I would think that the
chance of finding a copy of an 18-year-old check would be highly
unlikely. I once asked my bank if it could provide an old record, and
at that time I was told that the bank kept things on file for seven
years.

IF you still have your own bank statement showing the debit of the
check to your account, that would be evidence that the check was
cashed, presumably by the person or company to whom it was issued.
Since the statute of limitation for any claim that far back has been
exceeded, there can be no legal claim made now for the payment, thus
the assumption must be that the transaction was completed.

This is, of course, no legal or professional advice, as you can read
in the disclaimer below.