Well, it was only easy to convince my parents [with veganism] by gradually starting off with vegetarianism. The bad approach, at least in my experience, was starting off omnivore and immediately becoming a vegan. Vegetarianism alone was a very hard battle I eventually won by just saying, "Look, I'll rummage through the cabinets and make do with what I can, and if there's nothing
vegetarian, I guess I'll starve until there is."
Assuming you're already vegetarian, how about just casually depleting the eggs, then dairy products one at a time from your diet instead? You could try printing out some facts on vegan nutrition, and if it's still not convincing, look through the cabinets and see if you could make do with what's in there. Even if it's vegetable soup, crackers, peanut butter and jelly sandwiches, canned vegetables, whatever, it'll eventually get their attention and they'll most likely admire/detest your dedication enough to budge. You could also make some new suggestions on the grocery list while at it.
I told my mother I was vegan about four days ago when we stopped by Wendy's. After refusing to order a frosty, all she said was, "How long is the veganism going to last? You better not change your mind when we get home because I don't have enough gas to come back up here." and that was that. I guess it gets a little easier once you get past the vegetarianism route. Now, she actually believes I eat healthier than the rest of the family whereas before my diet was considered inadequate in nutrition and "very low" in protein.
A good site for vegan nutrition: http://www.vrg.org/nutshell/vegan.htm
EDIT: Well, it looks I had a major brainfart, so I'll just leave my original post for those of you who want to transition to veganism and are going through similar problems at home. :P