Is Eilat part of Eretz Yisrael?

Q: Is Eilat a part of Eretz Yisrael or outside of its borders?
A: Eilat is part of Eretz Yisrael for three reasons:
1. Eilat is mentioned in the Tanach and it is possible that it is part of the Land to be conquered. While it is not completely clear that it is part of this Land, it is certainly belongs to Eretz Yisrael.
2. Even if we say that it is not part of the Land to be conquered, it is part of the Land promised to us in the future. In this case, there is no obligation to conquer Eilat, but one who lives there fulfills the mitzvah of settling in Eretz Yisrael.
3. And if we say that Eilat is not part of the Land to be conquered and it is not part of the Land promised to us, the fact is that we did conquer it in the War of Independence. After all, the Rambam wrote in Hilchot Melachim (5:6) that when a King of Israel conquers land, that land is considered part of Eretz Israel. And Maran Ha-Rav Kook wrote that the State of Israel would be considered the quasi-Kingdom of Israel (Shut Mishpat Cohain p. 338).
Each of these reasons is enough to show that Eilat is part of Eretz Yisrael and all the more so when they are joined together (See Shut Tzitz Eliezer 3:23, responsa of Ha-Rav Ben Tzion Uziel ibid. and Ha-Rav Shaul Yisraeli in Eretz Chemdah vol. 1 sha'ar 1, 9).