Describe each of the following sets as the empty set, as R, or in interval notation, as appropriate: (a) igcap_{n=1}^{infty}left(-frac{1}{n}, frac{1}{n} ight)
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The set is given as the intersection of intervals \(\left(-\frac{1}{n}, \frac{1}{n}\right)\) for \(n\) ranging from 1 to infinity. Show moreā¦
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Theorem 1.4.10 [Nested Interval Theorem] Every nested sequence of non-empty closed intervals in ā has a non-empty intersection. To be more explicit, let (a_i) and (b_i) be sequences of real numbers satisfying the following properties: ā¢ For all i ā ā, a_i ā¤ b_i. ā¢ For all i ā ā, a_i ā¤ a_{i+1} and b_i ā„ b_{i+1}. Then ā©_{i=1}^{ā} [a_i, b_i] ā ā .
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