What is happening when a body part falls asleep is for a period of time it has been robbed of oxygen in the form of blood. So the blood has been restricted, and the nerves are robbed of oxygen as a result.
The nerves don’t hurt while we’re sitting on the body part because there’s no oxygen/blood being sent to the nerves. Then when we stand up or move around, the oxygen/blood starts to go back to the nerves and “wakes” the nerves up. The feeling we perceive is the nerves waking back up because they have oxygen again.
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