No, the HPV vaccine will not eliminate cervical cancer. HPV is a major cause of cervical cancer, so getting HPV shots is a good idea. Those shots will prevent the HPV virus, so they should prevent cervical cancer. However, the vaccine probably won't eliminate it once someone has it. It is only for preventing the disease in the first place.
No, HPV Vaccine will not eliminate cervical cancer. While the HPV Vaccine may help prevent cervical cancer, if it had eliminated it, we would already have the evidence to prove that it does. The HPV vaccine has been given out long enough to know what the vaccine does for people.
There are a lot of people who opt out of the HPV vaccine for their children. Unfortunately, until everyone is vaccinated, we will never completely eliminate cervical cancer. There is a lot of good that will come of the vaccine. Infection rates will slow, source of infection will become easier to detect, and people will be more aware of the cancer. But it will not stop HPV completely.
It is very rare that a vaccine completely eliminates a disease, and the makers of the HPV vaccine aren't claiming that their drug will either. It will greatly reduce the incidence of cervical cancer based on the evidence so far, but there will still be some people who for some reason don't react properly to the vaccine.