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Prove the result on a delta-net first

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In probabilistic arguments, one sometimes wants to prove that it is possible for a statement to be true for every element of an infinite set, and one knows that for each element of the set a slightly stronger statement is true with probability at least 1-\epsilon for some very small \epsilon. If there is a natural metric on the set, one can sometimes achieve this by proving that the set contains a \delta-net of size N<\epsilon^{-1}, and also that if the stronger result holds for every element of a \delta-net, then the weaker result holds everywhere. Techniques for proving the existence of small \delta-nets can be found in the article Finding small nets.

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