Shoot for the moon

You won't miss and definitely won't be among the stars. Set in late 22nd century A.D. we are constructing a space elevator that would physically connects Earth and the moon, the lift will be using electromagnets to ferry people between a airship floating in the sky and the lunar base station like a maglev train. Given that we have perfected asteroids mining technology and have near limitless resource to realize space elevator concept, how can we attach a cable that won't succumb to tension when the moon drift further away?

(Bonus) I'm thinking can a lift move along the cable using hall effect principle?

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Shoot for the moon

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