What you can do is show that person how easy it is to spoof the from address, something which can be very easily done in most e-mail software, including Outlook Express, Windows Mail, Thunderbird, etc. in much the same way as writing a letter in the post with someone else’s address as the return address.
Assuming you use software to read your e-mail (e.g. Outlook Express, Windows mail, etc.), you can try this yourself as an example as follows: Go into your e-mail account settings, change the field that contains your e-mail address field to "firstname.lastname@example.org" and change the your/display name field to “John Smith”. Now send yourself an e-mail and look what happens when it arrives. Don’t forget to change these fields back afterwards.
Unfortunately, as far as proving that you did not send a particular e-mail, this is more tricky and requires some analysis of the headers and understanding the workings. Follow mciahel’s link above.