The phrase "demonstrate perfectly" is correct and can be used in written English.
You could use it to indicate that something was completed very well, that the result was excellent, or that a skill was demonstrated without flaw. For example, "He demonstrated the technique perfectly and was praised for his effort."
Science writers like Jonah Lehrer have pointed to studies that seem to demonstrate perfectly respectable attention spans in gamers and Web users.
Palm oil and many other commodities like fossil fuels demonstrate perfectly the rising concern with a flaw in something around which the economy trustingly, and often unthinkingly, pivots – the price mechanism.
But before I get started, perhaps the Olympics demonstrate perfectly that mankind is not perfect.
So, don't expect Minerva to have classes that demonstrate "perfectly curated" diversity, don't expect it to have an NCAA Championship-winning football team, and don't expect to find any classmates who are there because their parents donated a new building.
In the Eremothecium genus, the single copies found in each species— E. gossypii, E. cymbalariae, and E. aceri are ancestral and surrounded by genes that demonstrate perfectly conserved synteny among the three species.
It also demonstrates perfectly why that data could not, of itself, be regarded as reliable evidence of blood doping.
Perhaps more than any other area of movie-making, animation demonstrates perfectly how the old and the new, past and future, can coexist.
Last week's Doctor Who Prom demonstrated perfectly the fine balance of the readily popular and the innovative that makes the Proms such a highlight of our cultural summer.
"This case demonstrates perfectly why our role is critical and that the banks are already using the office of the currency comptroller as a shield," Mr. Spitzer said.
Johnson's value was demonstrated perfectly in last Saturday's victory over the Denver Nuggets: he drilled three outside shots, including two 3-pointers, in the fourth quarter to lift the Knicks to victory.
Thomas Robert Malthus (1766 1834) demonstrated perfectly the propensity of each generation to overthrow the fondest schemes of the last when he published An Essay on the Principle of Population (1798), in which he painted the gloomiest picture imaginable of the human prospect.
