How can son prove he is only child?

The livret de famille is an administrative tool to simplify the rights of people married and/or born in France; no livret may be issued for someone born outside France unless he or she marries in France, but even then the details of the person's birth and parentage are included based only on what he/she declares and, as such, are not actual proof of anything. When you die, your son will instruct a notaire to settle your (French) estate; the notaire will require his full birth certificate as evidence that he is the presumptive heir and will require him to sign a statutory declaration to the effect that there are no other offspring. The existence or absence of a livret de famille is not important.