query: A compact set contains sup and inf
positive_passages.docid: 22131
positive_passages.text: Prove if nonempty $K \subset\mathbb{R}$ is compact, then both sup $K$ and inf $K$ exist and are elements of $K$.
negative_passages.docid: 125011
negative_passages.text: Show lim sup (an) = lim inf (an) for increasing for increasing sequence of subsets.
negative_passages.docid: 95234
negative_passages.text: Quasi-components in compact space
negative_passages.docid: 19289
negative_passages.text: complement of a set and its measure
negative_passages.docid: 41408
negative_passages.text: Which of the following sets are compact:
negative_passages.docid: 47810
negative_passages.text: $C(X)$ is separable when $X$ is compact?
negative_passages.docid: 63529
negative_passages.text: How to show that this set is compact in $\ell^2$
negative_passages.docid: 7112
negative_passages.text: Product of sets and supremum
negative_passages.docid: 18907
negative_passages.text: Is this proof of $\liminf E_k \subset \limsup E_k $ correct?
negative_passages.docid: 106957
negative_passages.text: Compact sets are closed?
negative_passages.docid: 81203
negative_passages.text: Supremum and infimum of sets
negative_passages.docid: 17212
negative_passages.text: The preimage of a subset
negative_passages.docid: 147213
negative_passages.text: Prove that finite unions of compact sets are compact
negative_passages.docid: 152817
negative_passages.text: Proving $\sup B=\inf A$
negative_passages.docid: 147013
negative_passages.text: Subset of a finite set is finite
negative_passages.docid: 29744
negative_passages.text: Compactness and axiomatisability
negative_passages.docid: 47820
negative_passages.text: How can I show a supremum of a set is also its limit point?