My friend and I had the following battle: I had 3 fighters and 5 infantry, and he had 3 infantry. At the end of the battle, all of my infantry got destroyed and all I had left was the 3 fighters. He lost all of his infantry. He said I couldn't take the territory because I didn't have any land units. I started the battle with land units but at the end of the battle, I didn't have any. What is the rule on this specific situation?
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The rulebook clearly covers this situation. On page 17 under Step 8: Capture Territory
You need to capture a territory with a land unit.
You have your choice of which units to sacrifice. Meaning that if you want the territory, you need to sacrifice more expensive sea and/or air units to preserve the last land unit.
Usually, the territory is not worth it. Then what you have achieved is destroying your opponent's units on the territory for the future.
The purpose of the so-called "infantry push mechanic" is to attack a territory with enough infantry so that you will have infantry left to take it over AFTER absorbing battle losses.