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:
“If any proof were needed of the progress of the cause for which I have worked, it is here tonight. The presence on the stage of these college women, and in the audience of all those college girls who will some day be the nation’s greatest strength, will tell their own story to the world.”
—Susan B. Anthony (1820–1906)
“The source of Pyrrhonism comes from failing to distinguish between a demonstration, a proof and a probability. A demonstration supposes that the contradictory idea is impossible; a proof of fact is where all the reasons lead to belief, without there being any pretext for doubt; a probability is where the reasons for belief are stronger than those for doubting.”
—Andrew Michael Ramsay (1686–1743)
“It comes to pass oft that a terrible oath, with a swaggering accent sharply twanged off, gives manhood more approbation than ever proof itself would have earned him.”
—William Shakespeare (1564–1616)