mth536_problem_7_9b

# mth536_problem_7_9b - MTH536 Chapter 7 Problem 9b Gene...

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

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: MTH536 Chapter 7 Problem 9b Gene Quinn – p. 1/ Chapter 7 Problem 9b Let be a point and a set in a metric space. Deﬁne £ ¡¦ £ ¤ ¥ © ¨ ¢  ¢ §    ¤   ¢ ¤ ¥ §  § ¡ ¦ ¡ Show that  £ ¡¦ – p. 2/ Proof of proposition Let . Suppose that £ ¦ ©¨  ! § "  ¢    ¤ ¥ ¡ £ Then by deﬁnition for any contradicting that there exists a £ ¤ ¥ ¦ ©¨  ¢ with #¦ ¢ ¤ , ! "  ¡ ! § so it must be that £ ¦ © ¨  ¢ ¤    §     # \$ ! – p. 3/ Proof of proposition Since was an arbitrary choice, it must be true that £ ¡¦ %  £ § ¡¦ ¡ &  ¢ ¤ ¥ §  ¡ ¢ ¤ ¥ and we conclude that   – p. 4/ Proof of proposition £ Let ¡¦ and let be given. Now £ ¦ ¢ ¤ ¥ £    § #¦ ¢ ¡  ¤ ¥  § &  ! "  ¢ ¤ ¥ §  implies that there exists a It follows that  ©¨ with ¡ . ¡ is a limit point of # and therefore and we conclude that  ¢£ ¡¦  ¤ ¡ \$ ! – p. 5/ Proof of proposition £ Let ¡¦ and let be given. Now £ ¦ ¢ ¤ ¥ £    § #¦ ¢ ¡  ¤ ¥  §  & ¡¦ §  § ¡  ! "  ¢ ¤ ¥ §  implies that there exists a It follows that  ©¨ with ¡ . ¡ is a limit point of # and therefore and we conclude that  ¢£ ¡¦  ¤ ¡ Together with the previous result this establishes that   ¤ ¢£ \$ ! – p. 5/ ...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online