Showing a metric space is bounded.

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This is from a review packet:

Let $d:\mathbb{R} \to \mathbb{R}$ be defined as $$d(x,y)=\frac{|x-y|}{1+|x-y|}.$$

i) Show that $(\mathbb{R},d)$ is a bounded metric space.
ii) Show that $A=[a,\infty)$ is a closed and bounded subset of $\mathbb{R}$.
iii) Show that $A=[1,\infty)$ is not compact.

For (i) I think this is a true observation: $d(x,y)\leq1$ for an arbitrary $x,y \in \mathbb{R}$. I'm not sure where to go from there however.

For (ii) and (iii) - I'm assuming those will following quickly from (i).

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1 Answer

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Since $d(x,y)\leq 1$ for $\forall x \neq y$ we get that $\mathbb R,[a,+\infty)\subset B_1(0)$. Finally $[1,+\infty)$ is not compact since the open cover $\{(0,n), \ n \in \mathbb{N}\}$ does not have a finite subcover of $[1,+\infty)$.

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