M25008PP1

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

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 09/24/2009 for the course MATH 250 taught by Professor Rogerhowe during the Fall '06 term at Yale.

Ask a homework question - tutors are online