absoulutely not!, the DS has wireless built in.
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No. There is no port on the Nintendo DS that supports the wireless adapter. Therefore, communicating with the Gameboy Advanced SP is impossible. You may argue that the DS itself has wireless capability and should link up without the adapter, but no. Games which use the wireless adapter in the SP cannot work with the DS (even in slot 2) wireless network. Try it. It simply asks you to insert the wireless adapter. And you can't do that on the DS.
The Nintendo USB Wi-Fi Connector is for people who want to play Nintendo DS games on a WiFi network but do not have a wireless modem. Just plug it into a computer that is connected to the internet and you can access the internet as a wireless network on your DS.
While you may have wireless functionality with the DS, all the mystery gifts were available during real life Nintendo Events which happened long before the DS was available, and therefore require the only wireless function available at that time.
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you cant