Convergence in the Hausdorff metric

[math8]

Let (X,d) be a metric space. Let {An} be a nested family of non empty compact subsets of X. Let A=Intersection of all An.
We have that A is non empty and compact.

We show An converges to A in the Hausdorff metric (D).

I know D(A,B)= Inf {t>or eq.0 : A C B_t and B C A_t} Where A_t is the t-parallel body of A meaning A_t={x in X: d(x,A) < or eq.t}.

But I am not sure how to proceed.

 

[Billy Bob]

Haven't thought this through completely, but some thoughts:

Let epsilon > 0.

Show there exists N such that $$A_N\subset A_{\epsilon}$$.

Suppose not. Get sequence of x_n in A_n but not in A_epsilon, then thin to convergent subsequence.

Obtain contradiction.