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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/ ...
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