It is impossible to actually sleep on the game, but you can rest at your house, the Team Galactic Nap Room (lame, right?), and certain inns around Sinnoh.
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There is no Tm sleep in diamond, there is a sleep talk, which you can find in eterna forest.
Yes. TM82 and is found in the eterna forest.
all the lava cookie does is heals any effect that is on your pokemon. sleep, poison, burn, etc.
Pokemon diamond
There is no TM yawn. Only certain Pokemon can learn yawn/hypnosis/sleep powder/etc.