By Dragoslav S. Mitrinovic, J. Pecaric, A.M Fink

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Definition 10. The Hausdorff distance d is defined as follows: for all elements P and Q of P(EN ), d(P, Q) = inf{r > 0 : P ⊂ Qr and Q ⊂ Pr }. 2 The Projection Map 49 Fig. , from l to m ). One has always |l − l | ≤ |mm | m’ l’ m l A One can easily check that d is effectively a distance on the set Pc (EN ) of compact subsets of EN . The main properties of dA are summarized in the following theorem. Theorem 10. Let A be a (nonempty) subset of EN . Then: 1. dA (m) = 0 ⇐⇒ m ∈ A. 2. The map dA : EN → R+ is uniformly Lipschitz and ∀m, m ∈ EN , dA (m) − dA (m ) ≤ d(m, m ).

1 Some authors simply use the term measure instead of signed measure. This can be done if no confusion is possible.

We shall prove that lim pn = p. n→∞ Since A is compact, there exists a subsequence of (pn )n∈N , which we still denote by (pn ), n ∈ N, which converges to a point q of A. Since the distance function dA is continuous, dA (mn ) tends to dA (m) = d(m, p). On the other hand, dA (mn ) = d(mn , A) = d(mn , pn ) tends to d(m, q). Hence, d(m, p) = d(m, q). Since m has a unique projection on A, p = q and lim pn = p. n→∞ This implies that all limiting values of the sequence (pn )n∈N equal p. Classically, we conclude that the sequence (pn )n∈N tends to p.