Prove that evolution has occurred/is occurring using examples.
There are actually many researchers who have observed evolution occurring in a variety of organisms. Keep in mind that large changes require long time periods, so the length of the average lab study limits the amount of change that can be documented. Probably the best know example is Belyaev's work with silver foxes; he showed that selecting for the trait of tameness created a complex of changes in a relatively small number of generations, including physical changes that had not been specifically selected for. Populations of stickleback fish have been observed to evolve to accommodate changes in environmental temperature. Italian wall lizards moved to an island showed dramatic changes to the structure of the gut, as well as behavioral changes, in just thirty years. Dozens of species of animals have shown changes in size, physical structure, and behavior because of hunting pressure from humans. These are just a few examples; this link will take you to an article with a chart summarizing many other examples.
Those who work with bacteria have seen faster evolution in the species they work with, because the reproduction rate is so rapid. The bacteria Wolbachia has been observed changing from a parasite to a mutalist, for example. Many bacteria, including streptococcus, have demonstrated evolution by rapidly developing antibiotic resistance.
Evolution by natural selection is a scientific theory, and as such cannot be proven beyond a shadow of a doubt, since there is always the possibility, however remote, that evidence could be uncovered to contradict it. However, no such evidence has ever been found, and an abundance of evidence uncovered over more than a century has supported the theory.
Gradual changes in form can be seen in the fossil record, indicating that evolution has taken place over millions of years. Many animals share structures that seem to point to a common ancestor, such as the pentadactyl limb. Similarities between different animals at the fetal stage also suggest some common ancestor from which many animals have evolved. In addition, the fact that a wide variety of animals share the same DNA structure, as well as certain structures within cells, is cited as strong evidence that species have evolved from other species.