Should I even give him a chance?

My bf has a long history of lying. In fact his entire family and mine have turned their backs on him for this reason. Usually it’s small like like saying he did chores he didn’t do. Well 4 months ago he pretended to have a job and didn’t so my family (the last people to give him a place to stay) kicked him out. He cheated on me and we were separated, he then asked me for money to book a flight to a friend. I know all of his friends and I knew immediately it wasn’t a friend so I told him no. She ultimately cancelled on him anyway. He was homeless one day and called me crying hysterically. So I let him come stay with me. But I asked him straight up where he was taking that flight to he lied, months passed and I found out the truth. Now here we are, we got past that but it’s been distant. I haven’t so much as held his hand since I found out that lie. Let’s just say, I’m not that attracted to a lying man-child. Now he’s a week away from his first paycheck at a new job (assuming that’s not also a lie) and I caught him lying again. This time he said he was going with a friend, walked out and walked back in and said they cancelled. I knew it was fake so I asked for proof THEN I watched him send him self a fake text pretending to be his friend and replying to himself and offered it as proof. It bewilders me why someone would make up a dumb lie like that. Should I give him until his paycheck or send his broke a— to a shelter.