How can I be sure what dealer told me is true

Actually 6 years is beyond normal battery life expectancies - they're more than likely being truthful.. When 'you' have the vehicle at home - you only use the battery briefly for starting - while at the shop being serviced - doors can be opened for awhile - diagnostic checks, etc will put a little more drain on the old and tired battery - then the failing battery is discovered. It's a common occurrence in the automobile repair business.