I don't know how to make him believe me that I will always be there for him
In my experience it helps to see this as part of the disorder, not him, if that makes sense.
Meaning, the content of what he says is disordered thinking. It's the disorder talking.
People with BPD have a hard time filling the emptiness they feel. So they look elsewhere to fill it because what is external is indistinguishable from internal. There are no boundaries.
It's more like he's saying, "I cannot feel myself. I feel empty. I am not whole."
Since that is a weird thing to know about one self, when one doesn't have a fully developed self, he looks for someone else to fill that void.
Which cannot be done.
This is why you read so much about validating someone who has BPD. We look for things we can validate (to a point). "You must feel so much pain" instead of "I don't know how many ways to assure you I'm here for you."
Validation must be paired with boundaries because the hole is deep and it's easy to disappear into the abyss if you prioritize his needs over your own.
Two helpful books (when it's siblings or child w/BPD) are Loving Someone with BPD by Shari Manning and Overcoming BPD by Valerie Porr. Both are often recommended by NEA-BPD Family Connections workshops.