query: Prove the set $K = \{\frac{1}{n} \mid n\in \mathbb{N}\}\cup \{0\}$ is compact.
positive_passages.docid: 26215
positive_passages.text: Proof Verification: Let $D=\{ \frac{1}{n} \mid n\in\mathbb{N} \} \cup\{0\} \subset \mathbb{E}^1$. Prove that $D$ is compact.
negative_passages.docid: 110635
negative_passages.text: Proving $\mathbb{N}^k$ is countable
negative_passages.docid: 48783
negative_passages.text: Prove that the sum of two compact sets in $\mathbb R^n$ is compact.
negative_passages.docid: 2882
negative_passages.text: Limit points of the set $\{\frac {\varphi(n) }n : n\in \mathbb{N}\}$
negative_passages.docid: 48162
negative_passages.text: Is the subset $[0, \sqrt2] ∩\mathbb{Q} ⊂ \mathbb{Q}$ closed, bounded, compact?
negative_passages.docid: 22464
negative_passages.text: Which of the following subsets of   $\mathbb{R}^2$ are compact
negative_passages.docid: 32630
negative_passages.text: Proving that a set in which any sequence having exactly one accumulation point converges is compact
negative_passages.docid: 140440
negative_passages.text: Proving $(A\times C)\cup(B\times D)\subseteq(A\cup B)\times(C\cup D)$
negative_passages.docid: 101486
negative_passages.text: Understanding that $A = \{x\in \ell_2: |x_n| \leq \frac{1}{n}, n = 1,2,...\}$ is compact in $l_2$.
negative_passages.docid: 146146
negative_passages.text: Proof that a closed subset of a compact space is a compact space
negative_passages.docid: 52135
negative_passages.text: Show that $A \setminus ( B \setminus C ) \equiv ( A \setminus B) \cup ( A \cap C )$
negative_passages.docid: 28391
negative_passages.text: Prove the compactness of a set B
negative_passages.docid: 18596
negative_passages.text: A set $A \subset l_1$ is compact if and only if closed, bounded, and one other condition
negative_passages.docid: 138607
negative_passages.text: Proof that a limit point compact metric space is compact.
negative_passages.docid: 11114
negative_passages.text: Is a closed subset of a compact set (which is a subset of a metric space $M$) compact?
negative_passages.docid: 36237
negative_passages.text: Closed set mapped to itself in a compact Hausdorff space
negative_passages.docid: 153573
negative_passages.text: Counter Example for the union of any number of compact sets is compact in T2-spaces