mth536_problem_7_11a

mth536_problem_7_11a - MTH536 Chapter 7 Problem 11a Gene...

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

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

Unformatted text preview: MTH536 Chapter 7 Problem 11a Gene Quinn – p. 1/ Chapter 7 Problem 11a Show that if is any metric on a set then is an equivalent metric for Prove that . § ¨¡ ¢© ¦ is a bounded metric space. That is,     ¤  §   ¡ § ¡ ¥¤ £ ¢ ¡ – p. 2/ Proof of proposition Let be an open set in . (There are always open sets because and are open in the metric topology). If is an arbitrary element, there exists a such that for every ,  § ¨¡ ©  ¦ ¡ !# "    $ %    & ' § !#   ¤     ¡ – p. 3/ Proof of proposition Let be an open set in . (There are always open sets because and are open in the metric topology). If is an arbitrary element, there exists a such that for every ,  § ¨¡ ©  ¦ ¡ !# "    $ %    & ' Let Let be given and suppose that § !#    Then for any ,   ¡ ¥ ¤     § §   & (# !# ¥ ¤ !# (# ) £ !#   is a value that makes (1) hold. !# ¥ ¤ !# ¢   & ' § (# £ ¤     ¡ – p. 3/ Proof of proposition All quantities are positive, so this simplifies to    ¥   0 1   0 1    & & ¥ ' '  !# !# !# § § § § !#   – p. 4/ Proof of proposition All quantities are positive, so this simplifies to    ¥   0 1   0 1    & & ¥ ' '     that !# !# !# § § § Since was an arbitrary choice, we can a guarantees when , so § for any   §¡ © ¦ § (#  &    2 ) 453  open in open in § ¨¡ ¢©  '  ¦ 67# !#   – p. 4/ Proof of proposition Now suppose is an open set in element, there exists a with &¦ (8 ¨¡ #§ ¢© &  ¤  . If is an arbitrary such that for every  , % ¢   & () # (# ' §   9     ¡ – p. 5/ Proof of proposition Now suppose is an open set in element, there exists a with &¦ (8 ¨¡ #§ ¢© &  ¤  . If is an arbitrary such that for every  , % ¢   & () # (# ' Let be given and suppose that hold. Let §      Note that by definition, for  & ¥ ¤ (# () # !# £ (#  % ¥ ¤    § §   £ ¥ ¤    § § B A  ¥¤ § £¤ A ¤ @  is a value that makes (2) , 9   ¢   ¤ § £ ¥¤  so and . ¤ (8 # ¤ & A !# $ % §  & ¤   ¡ – p. 5/ so for any § !#  ¡ ' § !# £ ' A § & (# – p. 6/    &  § £  ¢  ' with Proof of proposition Fur thermore, A §  &      ¥ ¤    §  §   ' § ¥ ¤ ¢ ¢  §  §   ¢  0   ¤ ¢ ¢  §  §   & !# ¢   § £ ¥ ¤ (# § ¥¤  ¤ ¢ §       § § !#  ¡ ' § !# £ ' A § (#  & '    – p. 6/ ¢     &  § £  ¢  ' with so for any Fur thermore, § (# (# A 0 ¢   ' ¢   & ¤ § ¢   & § A (# § &   0  Proof of proposition With the assumption that may write A  ¢  ¥ ¤    §  §      § ¥ ¤ ¢ ¢  §  §   § & (# ¥ ¤ (# § ¢   £ § ¤ ¢ §  ' ¢  0  ¥¤        ¤ ¢ ¢  §  §   & !# , all quantities are positive and we Proof of proposition Since and were arbitray choices,   §¡ ¢© ¦ open in open in Combining with an earlier result we have §¡ ¢© ¦ open in open in Since and define the same open sets, they are equivalent. ¢ § ¨¡ ©  C  ¦ § ¨¡ ©  '  ¦ – p. 7/ Proof of proposition Since and were arbitray choices,   §¡ ¢© ¦ open in open in Combining with an earlier result we have §¡ ¢© ¦ open in open in Since and define the same open sets, they are equivalent. In the course of the proof, we established that ¢ ¢  which means that § ¨¡ ¢© ¦ is a bounded metric space.  & ¤  §   ¡ § § ¨¡ ©  C  ¦ § ¨¡ ©  '  ¦ – p. 7/ ...
View Full Document

Ask a homework question - tutors are online