When the account is being checked, Jagex check the digital record of the account activity.
They can see if an item was on the account though they can't see more than a few weeks into the past.
Jagex have the digital record. There is no more proof you need to supply.
Lost item return is not a service provided unlimited times.
If you are having an issue with security, you would be advised to take a more stringent approach to your email, device and account security.
Work through this page of instructions:
Security tips
You don't need proof when Jagex has their own records of everything. We know they can see what items we had and the movements of them from at least 2 years ago.