This little writeup was originally a reply to a thread on an internet forum dealing with this passage.
Paul did not write the words of these chapters to prove election. Their purpose was to address the question of Israel's relationship to the gospel in this present age.
However, the flow of the argument related to one of his assertions seems to assume the idea of individual election. I will attempt to demonstrate that as we trace this assertion through the passage.
In 9:6, Paul asserts, "But it is not that the word of God has taken no effect. For they are not all Israel who are of Israel." As we come to the close of verse 13, however, he has not yet demonstrated his assertion. At that point, all he has demonstrated is that God's choices from Abraham on bring us to the choice of Jacob/Israel over Esau/Edom. The assertion that "they are not all Israel who are of Israel" has yet to be demonstrated, and he doesn't completely finish addressing that assertion until 11:10.
Through verse 13, Paul has dealt with the election of groups - tribes or nations. But in verses 14-29, he narrows it down to individuals. In Pharaoh, we see an example of one whom God hardened, and his allusion in 9:20-21 is to Isaiah 45:9-13, where God anticipates Israel's objection to His choosing a gentile leader (Cyrus) to be their deliverer from the Babylonian captivity. And after stating that there will be "vessels of mercy" who were "prepared beforehand for glory," called from among both Jews and Gentiles, he quotes some passages from Hosea and Isaiah to show that some - a remnant - will be called from among Israel.
9:30 through 10:21 describe Israel's present condition in that, as a nation, they have rejected the gospel, although the offer of salvation is open to any of them who believe.
In 11:1-10, we read that God has not "cast away" Israel. As it was in the days of Elijah, "at this present time there is a remnant according to the election of grace." Paul concludes his argument that "they are not all Israel who are of Israel" with "What then? Israel has not obtained what it seeks; but the elect have obtained it, and the rest were hardened." Those who have "obtained it" are apparently the "Israel of God" mentioned in Galatians 6:16, and it is apparently this "Israel of God" that he had in mind when he made the initial assertion that "they are not all Israel [i.e., the 'Israel of God'] who are of Israel [i.e., the nation of Israel].