Exercise 1.4.3

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Prove that $\bigcap_{n=1}^{\infty}(0,1 / n)=\emptyset$. Notice that this demonstrates that the intervals in the Nested Interval Property must be closed for the conclusion of the theorem to hold.

Ulisse Mini · Accepted Answer

Suppose $x \in \bigcap_{n=1}^\infty (0,1/n)$, then we have $0 < x < 1/n$ for all $n$, which is impossible by the archimedean property, In other words we can always set $n$ large enough that $x$ lies outside the interval.

Exercise 1.4.3

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Exercise 1.4.3

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