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The exercise is stated as follows:

Suppose that $\{K_i:i\in\mathbb{N}_{0}\}$ is a collection of nonempty bounded and closed subsets of $\mathbb{R}^n$ such that $K_{i+1}\subset K_i$, for $i\in \mathbb{N}_{0}$. Is $\bigcap\{K_i:i\in\mathbb{N}_{0}\}$ necessarily nonempty?

My answer would be "yes" based on the compactness of the considered subsets. Yet, something is nagging me by the way the problem is formulated, suggesting that there could be a counterexample, which I could not find.

I would be grateful if someone could confirm or refute the "yes",

Bufo Viridis
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  • In the related posts on this posts there are various dupes of this question. See e.g this or this – Marco Dec 20 '23 at 10:53
  • Your compactness argument is correct. – Paul Frost Dec 20 '23 at 10:53
  • As far as I can tell, the word "compact" is not used until p. 124. Since this is in the end-of-chapter Review Exercises and the wording doesn't ask for a proof (looking at other exercises in the book, I think when the author wants the reader to prove something, the reader will be told this explicitly), you're probably suppose to be able to deduce this as a straightforward consequence of previous results. In this case, what is being asked seems to be a very slight rewording of Exercise 6 on p. 77. FYI, this is the book I used (continued) – Dave L. Renfro Dec 20 '23 at 12:01
  • for my first topology course (an independent reading course in Spring 1977, directed by this person), and this 28 March 2006 sci.math post gives some of my thoughts about Kasriel's book. – Dave L. Renfro Dec 20 '23 at 12:01
  • Because the sets are bounded, they have infima. Because they are closed, the infima are each in the set. Since the sets are nested, the infima are increasing. Because the first set is bounded, the infima are bounded. Thus they converge. Where is that limit? – Paul Sinclair Dec 21 '23 at 15:43

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