Paul is pointing out that God did not reject the Israelites.
He used himself as an example.
He is a Jew, descended from the Benjamin tribe, who have converted into a Christian.
Paul used another example of Elijah.
Elijah was a prophet that existed during the days of King Ahab.
In 1 Kings 18:6 - 19:18, Elijah competed with the prophets of Baal to prove whose God is real.
Each party is supposed to set the bull sacrifice on the wood to be on fire by calling the name of God.
Baal prophets slashed themselves and call on the name of their God but nothing happens.
When Elijah prayed to God, fire burned up the sacrifice.
When Jezebel knew this, she wanted to capture and kill Elijah but he ran away.
Elijah ran away to Beersheba and rested under the broom tree.
The angel called him up twice to eat so that he has the strength to travel the journey.
Later, Elijah came to Horeb and went into the cave to rest.
When God asked Elijah what he is doing, Elijah complained to God that all the prophets have been killed except him.
God replied that there is still 7,000 prophets in Israel that did not worship Baal.
Just like Elijah's example, only a remnant of the Israelites will put their faith in Christ.
Not every Israelite will be saved, except those who have been called by God.
However, the Israelite is the first nation that God called to be his people.
The remnant of Israelites that came to put their faith in Christ are saved by the grace of God.
Salvation by grace is based on faith and not by law keeping.
The Israelites did not obtain salvation although they sought for it enthusiastically because they are stubborn.
Their stubbornness prevent from seeing the truth and accepting the gospel message.
Paul describe the Gentiles as a wild olive shoot that has been ingrafted into the tree.
The broken branches are the Israelites that did not accept the gospel message.
The olive tree represents the nation of Israel (Jeremiah 11:16).
Paul is trying to say that all Israel will be saved one day after the full number of Gentiles have come to believe in Jesus and get saved.
The Jews persecuted the Christians by putting them in prison and beating them during the days of Paul.
Despite that, God will never take back the promise he made to their ancestor Abraham.
God is not a person that will tell lie and he always keep his promises (Numbers 23:19).
Because the Israelites rejected the Messiah, a way has been opened for the Gentiles to get saved.
Jesus Christ was sent to the Jews first.
The Gentiles previously rejected God but they have been given salvation afterward.
Just like the Gentiles, the whole of the Jewish nation will become saved eventually after they rejected God first.