This idea is a bad, embarrasingly bad.

I once read that what prevent us from building space elevators, among other things, is that no alloy would withstand a collumn so high. 

The idea is to add to the bottom of the space elevator a series of lifting turbines. We would place them for as long as the air doesn't becomes too thin for the turbines to be effective.

The turbines would have to be somewhat special. They would need to be designed to be so redundant, that if one failed or was scheduled for maintenance, the elevator would not collapse. I believe this to be statistically possible to determine, but can't tell about the costs.