Sure, you can line them up such that the integers run out first. You can even line them up so the rationals line up first. There are an infinite number of ways to line them up. In order to satisfy the definition of them being the same size we only require that ONE of the ways of lining them up leads to them corresponding exactly.
It’s merely a definition, though. The kicker is that the definition is useful and consistent.
Sure, you can line them up such that the integers run out first. You can even line them up so the rationals line up first. There are an infinite number of ways to line them up. In order to satisfy the definition of them being the same size we only require that ONE of the ways of lining them up leads to them corresponding exactly.
It’s merely a definition, though. The kicker is that the definition is useful and consistent.