A "Service Book" is just a place where dealers stamp their mark and record the mileage for the key and major service schedules.
The same info is also on the bills/receipts for these services. So all that the "Service Book is, is a shorthand record of the overall work done.
IF the service history has all the bills, receipts for the vehicle and the work done on it and the vehicle reg. No., tell the buyer to Foxtrot-Oscar, he has all the proofs he needs, he is just being a pedantic ar$ehole.