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## Main Question or Discussion Point

I've been reading a book called Superfractals, and I'm having trouble with a particular proof:

The distance from a point [itex]x \in X[/itex] to a set [itex]B \in \mathbb{H}(X)[/itex] (where [itex]\mathbb{H}(X)[/itex] is the space of nonempty compact subsets of [itex]X[/itex] is:

[tex]D_B(x):=\mbox{min}\lbrace d(x,b):b \in B\rbrace[/tex]

The distance from [itex]A \in \mathbb{H}(X)[/itex] to [itex]B \in \mathbb{H}(X)[/itex] is:

[tex]D_B(A):=\mbox{max}\lbrace D_B(a):a \in A\rbrace[/tex]

for all [itex]A,B \in \mathbb{H}(X)[/itex].

The proof is to show that [itex]D_B(A) \leq D_B(C)+D_C(A)[/itex]. The proof goes:

[tex]\begin{array}{rcl}D_B(a) &=&\mbox{min}_{b \in B}d(a,b) \\

&\leq& \mbox{min}_{b \in B}(d(a,c)+d(c,b))\\

&=&d(a,c)+\mbox{min}_{b \in B}d(c,b)\\

\end{array}[/tex]

Then

[tex]D_B(a) \leq \mbox{min}_{c \in C}d(a,c)+\mbox{max}_{c \in C}\mbox{min}_{b \in B}d(c,b)[/tex]

How do we reach this last step?

**Definitions**:The distance from a point [itex]x \in X[/itex] to a set [itex]B \in \mathbb{H}(X)[/itex] (where [itex]\mathbb{H}(X)[/itex] is the space of nonempty compact subsets of [itex]X[/itex] is:

[tex]D_B(x):=\mbox{min}\lbrace d(x,b):b \in B\rbrace[/tex]

The distance from [itex]A \in \mathbb{H}(X)[/itex] to [itex]B \in \mathbb{H}(X)[/itex] is:

[tex]D_B(A):=\mbox{max}\lbrace D_B(a):a \in A\rbrace[/tex]

for all [itex]A,B \in \mathbb{H}(X)[/itex].

The proof is to show that [itex]D_B(A) \leq D_B(C)+D_C(A)[/itex]. The proof goes:

[tex]\begin{array}{rcl}D_B(a) &=&\mbox{min}_{b \in B}d(a,b) \\

&\leq& \mbox{min}_{b \in B}(d(a,c)+d(c,b))\\

&=&d(a,c)+\mbox{min}_{b \in B}d(c,b)\\

\end{array}[/tex]

Then

[tex]D_B(a) \leq \mbox{min}_{c \in C}d(a,c)+\mbox{max}_{c \in C}\mbox{min}_{b \in B}d(c,b)[/tex]

How do we reach this last step?