Yes, ' fall asleep ' is an idiom meaning ' begin to sleep '.
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AnonymousIt is not really wrong but idiomatic; it is not. These are:
Is this idiomatic?
I have to fight not falling asleep.
AnonymousI have to fight not falling asleep.The "not" here is strange -- this seems to mean that you want to sleep. If you mean that you try to stay awake, say "I have to fight (against) falling asleep."