M25008PP1

# M25008PP1 - Mathematics 250 Fall 2008 Proof Practice 1 MM...

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Mathematics 250 Fall 2008 Proof Practice # 1 MM, Exercise 2.5 of Chapter 2: Show that a closed ball B ( a ,r ) is indeed a closed set. Argument 1 : We ﬁrst argue directly in terms of the deﬁnition of closed. A set X is said to be closed if every sequence { x n } n =1 of points in X , that converges in R k , has its limit in X . So, let’s check this for X = B ( ~ a ,r ). Let { x n } n =1 be an arbitrary convergent sequence in B ( ~ a ,r ), and let the limit of { x n } n =1 be L . We want to show that L belongs to B ( ~ a ,r ); that is, we want to show that || ~ a - L || ≤ r . This will imply that B ( ~ a ,r ) is closed. Since { x n } n =1 converges to L , we can ﬁnd points x k that are arbitrarily close to L . That is, for every ± > 0, we can ﬁnd a k such that || x k - L || < ± . Also, since x k is in B ( ~ a ,r ), we know that || ~ a - x k || ≤ r . Using these facts and the Triangle Inequality, we see that || ~ a - L || ≤ || ~ a - x k || + || x k - L || ≤ r + || x k
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