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The climber is actually pulling downwards on the rope. S/he is trying to pull the rope down or out of the ceiling but cannot do so. If you think of the climber just hanging there the rope has a tension upwards to counter the weight of the climber. If you are to move up then equilibrium must be broken and the net force on the climber must be up so the rope pulls the climber upwards. Of course, this pull is to do with action and reaction but the effect is the same.

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Q: Why does a rope climber pull downward on the rope to move upward?
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