Suppose we relax the rules for the men, so that each unpaired man proposes to the next woman on his list at a time of his choice (some men might procrastinate for several days, while others might propose and get rejected several times in a single day). prove that this modification will not change what pairing the algorithm outputs.

The modification in rules to allow unpaired men to propose their own choice girl cannot affect the algorithm of pairing outputs.

Explanation:

As it is the 21st century, each and every woman knows her rights, good and bad. As men are only allowed to propose in this scenario, they can not compel women to be their partners. So it is totally up to women either proposed girl or woman goes to that man or not it is totally up to her. So this modification has no symbolic effect on the pairing algorithm.

you answered your own : -/