Well to be perfectly honest I don't think I had to "convince" them. See for the most part I am the one who shops for food, I am the one who cooks the food. So if I didn't buy it they couldn't eat it. They eat what I cook since I have always refused to be a short order cook. Ever since the kids were little we alway had this rule. Mom is the boss mom has always been the boss mom will always be the boss. Now that is a basically a generalization. When my son was sick and suffering from migraines he was getting into his teen years. Junk food vacuums would be a better description. Since he was so sick and in pain for the most part as long as I had things that could be grabbed and stuffed into the pie hole it was ok. I did feel for him on his bday. So I tried to make him a cake with no flour no eggs no sugar and no chocolate.

That cake looked good but tasted so bad the dog wouldn't even eat it!
I aslo have always had the kids and hubby if home involved with cooking. We work together on that. Maybe not every single meal ever single day but often. I made a different kind of bread that my son could eat. We had so much fun. When that first loaf came out of the oven we were all so excited. Of course what we ended up with was a brick. You could barley break off a piece with a hammer and chisel. So we took the rest of the dough and made some decretive items and xmass trees decorations. Needless to say we had great laughs.
I also had made some green soup. My son LOVED it! So did the rest of the family. So during all the flops that I had having something that was super good helped out a lot. Now years later and my son still makes that green soup.
Now it's just hubby and myself so things are different at least a bit. Getting hubby to change or to become more aware well that has been interesting. Still I do most of the cooking. We shop together. For the most part hubby is ok with eating any thing that is put in front of him. If all we have is homemade soup he is ok with that. If we just have a salad he is ok with that too. Of course the alternative is if he doesn't want to eat what I fix he is free to fix what he wants. Then he also knows if that happens then most likely I would just cook for myself and he would be stuck cooking all the time for him self

Again I am not a short order cook. If you don't like what I fix sorry go for it yourself.
We do go through times where I will get tired of cooking and it's his turn. Well at first it was alway pizza or mac n cheese and hotdogs.

So I just had a talk with him. I was very upset. Because he knows this food makes me ill. So why would he want me to be ill? I spend time cooking and planning and shopping. I work hard at it because I feel he is worth it. That's how I see love and that's how I feel loved. So I feel unloved and uncared for when all I get is a hotdog. I also feel like I'm unappreciated and worthless when I can put fourth so much effort then I receive crap food in return.
He felt bad and never really saw it that way. Then he explained that he just doesn't think about it. Is never sure what to fix. So we now have stuff on hand that he can make. He also has his old stand by his special egg dish. I told him what he was making was a frittata then he saw a chef making one on a cooking show. He was very pleased with himself. As he should be. He knew how to make something with a fancy name! We also from time to time will look up things on line like what is vit b12 good for? What foods have that in it? Then we have a contest to see who can name whats in the food we are eating. We also play name that spice / herb. We close our eyes and sniff the jars to see if we can name that spice/herb. We sniff taste and some times just dump a pinch on a paper towel so go by sight. We try new things. Learn new things. Sometimes I will only wear an apron when cooking.

That gets his attention really quick.
Now I am not perfect and I also have a hard time eating what makes me feel good. I am addicted to chips and bread. It can be hard to get back on the wagon when I fall off. I have tried the Paleo and I know it would be good for both of us. however after a few days or weeks I want some crap food so bad I can't sleep. aka 'withdrawals' This year is really bad since I didn't plant a garden.

Things just didn't work out that way. Maybe in time for fall / winter? Going to the garden to get "free" food is another great motivator.
So I guess to sum it up how I "convince" I'd say I don't. I play have fun educated experiment learn enjoy use repetition have humor and sarcasm . Which is usually in the grocery store when that frozen pizza or something just magically appears in the cart. Ok then so today what you are saying is we want to be broke poor fat miserable in pain have heart burn and die earlier? All righty then why don't you grab 2 or 3 more so we can go happy. Sometimes more is put into the cart and sometimes the one is taken out. When we do get more and have a royal pig out well that's when I go back to observation. No judgement just observe. I can't believe I ate that much it was so good oh man I feel bloated tired weak. How are you feeling honey. Then I usually get up and get desert. Several tums served on a plate. Usually that pig out fest is followed by hubby saying I don't think we should do that again. Once in a great while I just have to put my foot down. Believe me I have gotten strange disapproving looks in the store when I tell hubby I wont even have that crap in my cart touching my good food no put it back!
Sad thing is when you have Drs media and the masses telling you you need to make changes slowly. People use that as an excuse to go to slow. Really go slow? Wow if you have a heart attack do you want them to go slow? What if you got cancer would you want to "go slow"? How bout you make the changes fast in other words just do it? Go through the detox and come out the other side feeling better vs going slow and then ending up in the hospital wanting to feel better fast.
Also I think a part of the education process is observation in the grocery store. Looking for the lies and how we are duped into believing what can't possibly be true. Like organic mac n cheese in a box. Really how can that possibly be organic? Or better yet the "Whole" grain scam. If it was truly Whole we wouldn't be able to chew it!! Don't believe me go out into a field pick a grain off a plant does it look the same as you find in a box? Try and chew it. Is it soft and chewy like the kind you find in the package? Lower in sugar so it's good for you. Oh by all means 3/4 cup a sugar will have you in tip top shape so fast vs 1cup sugar. But hey they put in only 1/2 the sugar

great but twice the corn syrup. My all time fav is when they put pictures on the box of some buff athlete or some tall thin perfectly balanced woman. The question is how many bars boxes or packages do I have to consume before I will look like that? People that look like that eat way different and work out hours and hours a day. I don't care what they say I will never be a young tall thin blond. Reality check I am a 5' 50yr old graying brunette Oh please give me a break

So there's my rambling followed by my rant. Hope it helps or at least gives you a chuckle or two.
PS Don't ever give your hubby powdered hot peppers that you encapsulate into capsules. OH he was not happy with me. Gave a whole new meaning to hot crossed buns.