1. An operation performing system, comprising: a device connected to the Internet via a router and capable of performing a certain operation; a service providing server on the Internet configured to provide a service that causes the device to perform the operation; and a relay server having a first storage and configured to relay between the service providing server and the device, wherein the device and the relay server are connected in a server-push enabled state in which the relay server transmits information generated on the Internet to the device via the router by use of a communication protocol and the device subsequently receives the information transmitted from the relay server, wherein the service providing server is configured to execute a first command-transmission processing for transmitting a performance command that commands performance of the certain operation and that includes specific information for identifying operation information representing the operation to be performed by the device, wherein the relay server is configured to execute a first reception processing for receiving the performance command transmitted from the service providing server, a second command-transmission processing for transmitting, in the server-push enabled state, the received performance command including the specific information to the device by use of the communication protocol, and a first storage processing for storing, in the first storage, service identifying information that identifies the service providing server, in response to reception of the performance command transmitted from the service providing server, wherein the device is configured to execute processings comprising a second reception processing for receiving the performance command transmitted from the relay server in the server-push enabled state, an acquiring processing for acquiring the operation information identified by the specific information included in the received performance command, an operation performing processing for performing the operation referring to the acquired operation information, and a first status-transmission processing for transmitting, to the relay server, status information indicative of a status of the operation that is being performed in the operation performing processing, when timing to transmit the status information comes during performance of the operation, and wherein the relay server is configured to execute a third reception processing for receiving the status information transmitted from the device, a second storage processing for storing in, the first storage, the received status information in association with the service identifying information, and a second status-transmission processing for transmitting, to the service providing server that is associated with the status information, the status information stored in the first storage in the second storage processing and not yet transmitted to the service providing server; wherein the device has a nonvolatile second storage, wherein the device is configured to execute a third storage processing for storing, in the second storage, the status information indicative of the status of the operation that is being performed in the operation performing processing, every time when timing to store the status information in the second storage comes during performance of the operation, and wherein, in the first status-transmission processing executed by the device, where not-yet-transmitted status information is stored in the second storage, notification information is transmitted to the relay server, together with the not-yet-transmitted status information, in response to switching of the power source of the device from off to on.