If i bring my router from my old apartment to a new one, am I still connected to my old internet service?

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I think what confuses you is the fact that your router is sending you a signal making you think you're getting an internet connection. That's not true. Routers do this regardless of whether you have internet signal coming in or not. So no, you're still using your neighbors' internet. If you needed to set up your internet connection with the same company they'd have to actually install it outside of your apartment on those tall poles with wires out on the street (idk what it's called sorry).