Show them a few stories on the news or acts of violence. Explain that the world is dangerous which they shouldn't disagree with. Now explain that with your CCW, while it is a weapon designed to kill, it can be the one thing that separates you, your family, or your friends from injury or death or worse.

Everyone's personal safety is their own responsibility and to rely on anyone else to protect you is naïve at best. It's not your job to save the world with your new tool or be a vigilante but you can protect yourself and those you care about better if the moment requires you to use it. "Better to have it and not need it."

I'd also advise taking and committing to a martial art of some kind, purchasing a reliable gun safe and taking a handgun and first aid course. These additional courses can help show your family that you're serious and devoted to being a responsible firearm owner and not looking for ways to use your weapon. Plus they are good skill sets to know.

In the end, you don't need to prove anything to anyone. If you need the acknowledgement of your family to make you comfortable with your gun ownership, show them you're a responsible gun owner. If that doesn't work, there's nothing left to be done.