How does anti-biotic resistance occur?

Anti-biotic resistance is a growing problem for humans now, exacerbated by the overuse and over reliance on anti-biotics. Bacteria colonies have genetic variation amongst the individuals which means they react differently to the anti-biotic, some may have resistance. When anti-biotics are used, the bacteria without resistance die, leaving a few  alive, mostly the resistant ones remaining but with some lucky non resistant ones too. As they now have much more resources such as food and space, they can reproduce at a faster rate and the population now has a higher proportion of the bacteria with resistant alleles. With continued anti-biotic use, the allele frequency for the resistant allele will increase until all bacteria contain it.