- Definition: Each set of restricted real numbers has a supremum. If a set A is limited to bottom then a -A assembly is limited to the top. According to the above definition, -A soup exists. As a result
inf A = – A -sup.
This means that a limited set is down to a minimum. - Archimedes properties
- For each real number x there is the original number n with x <n.
- For every x> 0 there is a natural number n with x <n.
- For every x> 0, there exists a single original number n such that n-1≤x<n
- Example 1: Zero is the infinity of a set
A = {1, 1/2, 1/3,…} = {1 / n / n = 1, 2, 3,…}
proof =
For every a = 1/nϵA there is a> 0. So zero is the lower bound of the set A.
Take any number e> 0. According to Archimedes’ properties, there is an original number of n with 1 / e <n. Earned (1 / e) <n or 1 / n <e = 0 + e.
then 0 is infinity A. - Example 2: If for every x> a implies x> b, prove that a≥b.
Answer =
The set of A = {x / x> a} = (a, infinity). It is easy to prove that a = inf A.
Take any xϵA. As we know, x> b is obtained. So b is the lower bound of a set A. Given the infinity definition, we obtain b≤a.
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