(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Given a compact set A[itex]\subset[/itex][itex]\Re[/itex][itex]^{n}[/itex] and a point x[itex]\in[/itex][itex]\Re[/itex][itex]^{n}[/itex] define the distance from x to A as the quantity:

d(x, A)=inf({[itex]\left\|[/itex]x-y[itex]\right\|[/itex]: y[itex]\in[/itex]A})

Given two compact sets A, B [itex]\subset[/itex][itex]\Re[/itex][itex]^{n}[/itex], define the Hausdorff distance between them to be:

d(A, B)=max(sup{d(x, B) : x[itex]\in[/itex]A}, sup{d(x, A) : x[itex]\in[/itex]B})

a. For any compact set A, prove that the function f : [itex]\Re[/itex][itex]^{n}[/itex][itex]\rightarrow[/itex][itex]\Re[/itex] given by: f(x)=d(x, A) is continuous.

b. For any two compact sets A, B it's true that: d(A, B)<[itex]\infty[/itex]

c. For any two compact sets A, B it's true that d(A, B)=0 if and only if A=B.

2. Relevant equations

3. The attempt at a solution

I think I handled part a. I'm just completely lost on b and c. Any help?

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# Homework Help: Hausdorff Distance Proofs

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