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First example: Minimal positive number

Problem without solution This selebrated problem dates back to the notorious Zenon and the Greeks. It sounds innocent
Find the minimal positive number.
This problem has no solution since a half of any positive number is smaller than this number and still positive!

Solution to the problem without solution

The ``solution'' was found (or defined) much later using the idea of limits. Limits of a sequence of elements of a set may not belong to that set but then they belong to its boundary. These limits can be added to the set. This operation is called closure and the set with its limiting elements is called closed.

In our example, the limit of a minimizing sequence of positive numbers is zero, which is not a positive number itself.

Here, the set of positive numbers is expanded by one element - zero.

Andre Cherkaev