# 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

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 $\frac{X}{{2}^{N-1}}$ is less than epsilon
2) Topology: Let Z be an open subset of a Topology X. Prove that $\int \left(Z\right)=Z$
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Step 1
Given:
We know that

Now
Therefore, by archimedian property $\mathrm{\exists }{N}_{0}ϵ\mathbb{N}$
Such that