Question: TRUE-FALSE. If your answer is "True," you don't need to prove it, just carefully sketch by quoting a definition or theorem or state a reason. If your answer is "False", please give a counter-example.

1. Suppose that \(S \subset \mathbf{R}\) contains infinitely many points, then \(S^{\prime} \neq \emptyset\). where \(S^{\prime}\) is the set of all accumulation points.
2. \(S \subset \mathbf{R}\). If \(s=\sup S\), then \(s \in S\).
3. \(S \subset \mathbf{R}\). If \(s=\sup S\), then \(s\) is an accumulation point of \(S\).
4. \(S \subset \mathbf{R}\). If \(S\) is compact and \(T \subset S\), then \(T\) is bounded.
5. If \(\left\{s_{n}\right\}\) is a sequence of real numbers and for some \(k \geq 1, \lim _{n}\left(s_{n+k}-\right.\) \(\left.s_{n}\right)=0 .\) Then \(s_{n}\) is a Cauchy sequence.
6. If a set contains no interior point, it is closed.
7. If \(s_{n} \rightarrow s\) and \(a
8. If \(\left\{a_{n}\right\}\) is a bounded monotone sequence. then \(\left\{a_{n}\right\}\) is a Cauchy sequence.

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