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Problem 46: (Practice With Neighborhoods)

Consider the interval $S=(1,8)$.

  1. Find real numbers $x$ and $\varepsilon$ so that $S=N_\varepsilon(x)$.
  2. Find real numbers $x$ and $\varepsilon$ so that $N_\varepsilon(x)$ is a proper subset of $S$.
  3. For every $x\in S$, give a formula for $\varepsilon$ so that $N_\varepsilon(x)\subseteq S$. Your choice of $\varepsilon$ will depend on $x$.

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