Of course if the Arabs did not want the Jews to build in the old city, they could just fail to sell them the land. Jews are not allowed to build without permits, and without a deed to the land they cannot get permits.
All uncomfortable litte details the Arabs and the Media ignore as it gets in the way of their common agenda.
Something that you may have noticed is that the entire neighborhood of Beit Yisrael is several feet below the level of Mea Shearim. This is because the land Beit Yisrael is built on was once a malaria-infested swamp called Bitzaat Hamavet (the "Swamp of Death"). Jews drained the swamp, cleared the land, and built on it.
I used to live in Beit Yisrael about two blocks away from what used to be the Jordanian border.