Air-conditioner communication system, connection confirmation method for air-conditioner communication system, wireless LAN adapter, and connection confirmation program for air-conditioner communication system

This air-conditioner communication system is provided with: an air-conditioner; a wireless LAN adapter which is provided as an attachment to the air-conditioner and has the function as a network communication unit for the air-conditioner; a router which, together with the wireless LAN adapter, configures a local area network; a server that can communicate with the router via a telecommunication line; and a terminal device that can communicate with the wireless LAN adapter and includes a display. The wireless LAN adapter acquires connection information indicating whether communication with the router is possible, and has the function of providing the connection information to the terminal device.