Apologies, as I think you are not a native english speaker, but your post is "word salad".
first sentence ... "Some scientists have an idea to build an elevator that could pull the spaceship from the ground of the earth to the spaceship orbiting around the earth." This is quite clearly talking about the space elevator concept, where a cable is erected to carry "stuff" up the geo-stationary orbit, and to balance the cable a counter weight is added above geo-stationary orbit so that the weight of the whole is balanced by the centrifugal force of the counterweight. And yes, this is a concept but some research is going on the see if we can make it a reality. Google "space elevator".
Then 2nd sentence "However, during the mission, the sling that is used to pull the spaceship from the ground to the spaceship orbiting around the earth is broken." Sling? where did this come from ? if it lifts the vehicle, what does it pull against ? I think this is not the right word, and so confusing the replies. The current idea is for the lift vehicle to be powered, rather than lifted, possibly by solar cells with the energy beamed to the lift vehicle by ultrawaves.
Because you then go back to the space elevator concept, the key problem is the strength to weight ratio of current materials. We need more material to react the load, but this extra material causes extra load, etc. Yes, this is the key design problem with space elevators. Currently the best material we have are carbon nano-tubes. Such a space elevator is, in my mind, built starting at geo-stationary orbit and building in both directions, keeping the structure balanced.
Building a space elevator on Earth is a very daunting, difficult, task. I suspect we are more likely to see one on the Moon or even Mars before we see one on Earth
"Wir hoffen, dass dieses Mal alles gut gehen wird!"
General Paulus, Nov 1942, outside Stalingrad after the launch of Operation Uranus.