NOTE: Archimedean Principle
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The Archimedean principle is: If and are real numbers with , then there exists a natural number such that . So , by this we can have a…NOTE: Archimedean Principle
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Programming Language Theory • System Programming
NOTE: Archimedean Principle
The Archimedean principle is: If aaa and bbb are real numbers with a>0a > 0a>0, then there exists a natural number nnn such that na>bna > bna>b.
So n>ban > \frac{b}{a}n>ab, by this we can have a particular example.
Let a=ϵa = \epsilona=ϵ and b=1b = 1b=1, then n>1ϵn > \frac{1}{\epsilon}n>ϵ1, and ϵ>1n\epsilon > \frac{1}{n}ϵ>n1.
Example
Show that inf({1n:n∈N})=0\inf \Big(\Big\{ \frac{1}{n} : n \in \mathbb{N} \Big\} \Big) = 0inf({n1:n∈N})=0.
Proof
Let A={1n:n∈N}A = \{\frac{1}{n} : n \in \mathbb{N}\}A={n1:n∈N}. Since 111 and nnn are positive for each n∈Nn \in \mathbb{N}n∈N, shows 1n>0\frac{1}{n} > 0n1>0, so 000 is a lower bound of AAA.
Let ϵ>0\epsilon > 0ϵ>0, by Archimedean principle there exists some n∈Nn \in \mathbb{N}n∈N such that 1n<ϵ\frac{1}{n} < \epsilonn1<ϵ. This element is in AAA and is less than 0+ϵ0 + \epsilon0+ϵ. Thus, 000 is infimum of AAA by definition: For all ϵ>0\epsilon > 0ϵ>0, 0+ϵ0 + \epsilon0+ϵ is not a lower bound of AAA.
author: Lîm Tsú-thuàn/林子篆/Danny
category:math
tag:notereal analysisarchimedean principle
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