Are both natural selection and evolution proven?
Natural selection is a natural process of genotypic and phenotypic selection, for an edge at survival.
Natural Selection is simply the selection of a hereditary characteristic and passing it on from one generation to another, which might or might not prove to be beneficial for the survival of an organism. (Darwin and Wallace first theorized it.)
If a hereditary trait is beneficial for the survival of the species, it is retained and passed on to the future generations and if not, the trait is removed gradually in the future generations. Sometimes, the trait unfit for survival could be removed as the whole species could also be wiped off the gene pool, thus resulting natural extinction of that particular species.
Natural Selection acts as the primary tool for evolution. As Darwin and Wallace observed, these natural selection of traits essential for survival, gradually over generations, tend to change the genotypic and the phenotypic makeups of the individuals of a certain species. This could also lead to the emergence of new species from the existing ones. This is the theory for evolution, based on Natural Selection.
Example of evolution via natural selection of traits : Disappearance of tail-bones in humans over time, Shortening of pinna, Decrease in the human head size (compaction) etc.