I have a few friends who suffer from anxiety as well and it took me a while to really understand this as well.
The short answer is, you can not convince them. The best you can do is to show them you like them, again and again. (II mean in a normal, day to day-to-day friend way. Not like, not by saying every few moments you like them every few moments.)
We all are a little anxious from time to time, but to them it is much more severe. They struggle with a continouscontinuous doubt that what they are doing is bad in somewaysome way or that they might offend you and others by doing seemingly mundane things to the rest.
Because of that, it is very hard for them to hold on to this feeling/ emotionemotion of "This person actually likes me". They can succeed in that, but know that on a bad day, they will doubt it all again.
From personal experience, I found that the best watway to handle it is to let them know you understand that they have this doubt every now and then. And that you will be there for them when they need you. Aside offrom that, just be a good friend.
I found that trying to help by suggesting things or urging them in a certain way to change about it is a bad move. It is very hard for them to change this and it will feel as if they are not okay or good enough for you and only make things only worse.