Prove that in a bit string the string 0 1 occurs at most one

Prove that in a bit string, the string 0 1 occurs at most one more time than the string 1 0.

SO - I understand that if the string begins with a 0 and ends eith a 0 or starts and ends with a 1, there will be an equal number of 01\'s and 10\'s. And that if it starts with a 0 but ends with a 1, there will be one more 01 than 10; if it starts with a 1 and ends with a 0, there will be one more 10 than 01.

However, I am not sure how to write up this proof correctly. It may appear on my test, and my professor is a motoriously harsh grader for proof questions, so I would greatly appreciate any help.

Solution