Question

When a bacterial infection is treated with an antibiotic, the hope is that the antibiotic will kill all of the bacteria and prevent them from reproducing. However, if the full prescription of antibiotics is not taken by the sick person, some of the more resistant bacteria can survive. When they reproduce, these bacteria can pass the genes responsible for their resistance on to their offspring. As a result, subsequent administration of the same antibiotic will be less and less effective over time.

Answer 1

The answer is true When a bacterial infection isn't treated fully then it would become harder to treat 

Answer 2

This is natural selection. If a random mutation/variation occurs and a bacteria is now resistant to antibiotics, then this is a favorable gene that promotes survival in its environment. This gene will be passed on to offspring through mitosis/binary fission or even conjugation and those bacteria will also be resistant. Eventually all non resistant bacteria will be gone or their numbers will be greatly reduced and only resistant bacteria will survive to reproduce.