Yeah, note that Nintendo DS / Lite / DSi are all the same software.
The only difference is that DSi has more features, and cannot play Gameboy Advance games.
Yes you will be able to play it on both DSi and normal DS.
How to change your username
How to add a user avatar
Why was my post hidden?
The points system
Flags vs Downvotes
Type a Pokemon name, type, move or ability to see instant results.