Proof
Let . For every input, we have . Therefore, . Using Fubini's theorem, since all terms are non-negative we can switch the order of summation, giving . By the Pigeonhole principle, there must exist an algorithm so that, concluding the proof.
As mentioned above, this theorem can also be seen as a very special case of the Minimax theorem.
Read more about this topic: Yao's Principle
Famous quotes containing the word proof:
“Ah! I have penetrated to those meadows on the morning of many a first spring day, jumping from hummock to hummock, from willow root to willow root, when the wild river valley and the woods were bathed in so pure and bright a light as would have waked the dead, if they had been slumbering in their graves, as some suppose. There needs no stronger proof of immortality. All things must live in such a light. O Death, where was thy sting? O Grave, where was thy victory, then?”
—Henry David Thoreau (1817–1862)
“War is a beastly business, it is true, but one proof we are human is our ability to learn, even from it, how better to exist.”
—M.F.K. Fisher (1908–1992)
“From whichever angle one looks at it, the application of racial theories remains a striking proof of the lowered demands of public opinion upon the purity of critical judgment.”
—Johan Huizinga (1872–1945)