Deleted text in red / Inserted text in green

WW

HEADERS_END

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.

[[[>50 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.