I really like the definition I heard once in a talk distinguishing "grace" from "mercy": Grace is a gift from God we don't deserve and mercy is a punishment we deserve yet He chooses to not inflict on us. So the remnant of Israel that Paul is talking about and the remnant of God's people that we want to be members of today are those blessed with a special gift they/we did not deserve. Since we didn't merit the gift when we received it, we can hold onto hope that it will be offered and received by every individual we encounter. Every hopeless case, whether family, friend, neighbor, politician, public sinner, or private offender, is a potential convert to the Lord. His mercy is abundant enough to forgive every wrongdoing and His grace is available although we had no hope of deserving it. So in every situation we can be thankful to Him for His gifts and hold onto hope for those who have not yet received them.