Cauchy Sequence

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Think of a sequence of dots on a plane (or line). Let's suppose that they have the following property:

You give me a coin.
I can find a place to put the coin such that

all but finitely many points are covered.

This works no matter how small the coin is.

In effect, we are asking that the points approach a limit. The reason for stating it as we have is because the limit point might not be in the set itself.

Such a sequence is called a Cauchy Sequence.

Here are some examples on a line:

1/2, 3/4, 7/8, 15/16, 31/32, ...
1, 3/2, 7/5, 17/12, 41/29, 99/70

Here are some sequences that are not Cauchy Sequences:

1, 2, 3, 1, 2, 3, 1, 2, ...
1, 2, 3, 4, 5, 6, 7, 8, ...
1, -1, 1, -1, 1, -1, ...

Cauchy sequences are a method used to construct the real numbers from the rational numbers.

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