1) Let X and epsilon > 0 be given. We need to use the Archimedian Principle to prove that a Natural Number N exists such that X/(2^(N-1)) is less than epsilon; 2) Topology: Let Z be an open subset of a Topology X. Prove that Int(Z) = Z

Alexandra Richardson 2022-07-15 Answered
1) Let X and ϵ > 0 be given. We need to use the Archimedian Principle to prove that a Natural Number N exists such that X 2 N 1 is less than epsilon
2) Topology: Let Z be an open subset of a Topology X. Prove that ( Z ) = Z
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Answers (1)

yatangije62
Answered 2022-07-16 Author has 16 answers
Step 1
Given: x > 0 ,   ϵ > 0
We know that
2 n 1 > n ,   n 2 1 2 n 1 < 1 n ,   n 2 x 2 n 1 < x n ,   n 2 ( x > 0 ) ( 1 )
Now x > 0 ,   ϵ > 0 ϵ x > 0
Therefore, by archimedian property N 0 ϵ N
Such that
1 n < ϵ x n N o ( 2 ) ( 1 ) + ( 2 )   gives x 2 n 1 < x n = x ( ϵ x ) < ϵ ,   n max { N o ,   2 }   N   ϵ   N   Such that   N max { N o ,   2 } Such that x 2 n 1 < ϵ
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