I'm going to work on a project without my friend
Why do you want to work alone?
I feel I need to prove I can do things on my own.
While that's admirable, you should also make sure to not leave your friend out on the lurch. Just be cognizant of that and you'll be fine.
I'm going to do what I think is best and make sure to not get my friend involved in this.
I am proud of you for taking the initiative. It can be really difficult to do things on your own sometimes. Please reach out if you feel like you are falling behind.