WordPress, HTML5 and Flash

I am in contact with someone who has an existing site that manually requires the user to either select a Flash Site or HTML5 depending on the device used to view the site. I have suggested a wordpress site would be much more convenient for him and will provide easily maintainable site that will be responsive to the device (I have selected a responsive theme) to the device. He is still insistent that he will need two sites.

My intuition is that I do not think he needs to worry about separate sites for different devices, however I do not have the vocabulary to describe why he does not need the two sites. Can anybody help with some words to explain.