What will happen if I delete account from PS4 ?

I've recently bought an used PS4. On it there was an account from a guy I bought it from, with 8 games (pretty good games too). When I was buying it, we didn't agree on buying his PSN account too, we agreed on buying a console only, so when I didn't manage to log in onto PS network with his account because he changed the password I wasn't surprised. However, when I logged in with my own account, I was able to play the games that HE owns. So, my question is is this allowed or am I risking a ban from PSN for playing games that I don't own ? What if his account gets banned, am I risking a ban on my console/ my account ? And finally, if I decide to delete his account from the console entirely, will I still keep the games and be able to play them ?