How do we prove that we are improving the continual improvement process
I think, the act(s) of Improvement is a process. It has inputs in form of methods, procedures etc. resulting in issues, problems or simply opportunities for improvement. It has outputs in form of better procedures, operations, processes etc. The methods, techniques, approaches we use for converting inputs into outputs is the process of 'improvement'.
Now, the degree of success in converting inputs to outputs can be assessed for the efectiveness of the methods we used. Examples are reduced customer complaints, shorter cycle time etc.
The other measure is the effeiciency i.e. using less resources and achieving better results. The improved efficiency may be due to better methods, for example, for measuring customer perception and associated QFD, team approach towards improvement indicated or measured through employee suggestions program, better auditing (process) approach and follow up methods etc.
The above explanation can be supported with several practical examples that I have not got into for keeping my note short.