Unformatted text preview: MTH536 Chapter 7 Problem 1a
Gene Quinn – p. 1/ Chapter 7 Problem 1a
Show that the set of all tuples of real numbers becomes a metric space under each of the following metrics:
¢ ¤ ¥£ ¨ ¤ § ¤ ¦ § § # ¦ ¦" " " ¦ ¡ § © ¤£ ¨ ¤ ! § © ¤ ¡ § – p. 2/ Proof of Proposition
Any metric must satisfy the following four conditions:
¨ ¡ ¤ ¥£ $ ¨ ¦ § % ¨ & § ( ¤ & © ¦ ¨ £ ) § ¡ ¦ ¨ © ¤£ ¨ ¨ ¤ ¥£ ¦ §¤ © ¨ ¡ ¤ ¥£ ¨ £ 0 1 ¡ § ¡ 2 ¡ ¦ ¦ ¦2 § § ¡ ¨ ' ¡ ¤ ¥£ ¨ – p. 3/ Proof that
Since 4 and satisfy condition 1
¨ ¤ § ¤ § § © 3 ¢ ¤£ ¡ 3 is a sum of nonnegative reals for all ¦ , is always . 6 ¢ ¤ § 5 ¦ ¡ % & – p. 4/ Proof that
Since 4 and satisfy condition 1
¨ ¤ § ¤ § § © 3 ¢ ¤£ ¡ 3 is a sum of nonnegative reals for all
¤£ ¨ ¤ ! ¦ , is always
# . 6 ¢ ¤ § 5 ¦ ¤ ¡ § § ¡ © ¦ ¦" " " is the maximum of a ﬁnite set of nonnegative real numbers, so for all , is always .
6 § 5 ¦ ¡ % & ¤ ¦ § % & – p. 4/ Proof that 4 and
§ © satisfy condition 2
, and
$ 1 § 9 1 7© $ 1 9 ¦ 1 87¤ 3 Suppose . Then by deﬁnition,
¤7 § 7 © ¤ 3 ¦& – p. 5/ Proof that and
¤ © § ¡ ¤ ¥£ ¦ § ¨ ¦ § ¨ 7 ¤7 @ 7 § © ¦" & " " ¦ & © & © & 1 @ & 7 © 1 9 1 $ © ¤7 © 7 § 1 9 7 § ¦& $ ¦ © 7 ¡ ¢ ¤£ ¤7 Suppose It follows that . Then by deﬁnition,
87¤ 7© § 3 4 and
3 satisfy condition 2
, and – p. 5/ Proof that 4 and
§ © satisfy condition 2
, and
$ 1 § 9 1 7© $ 1 9 ¦ 1 & § © 7 7 ¦" & @ © ¦ 7 & § © 7 " " ¡ ¢ ¦ © & & 87¤ 3 Suppose . Then by deﬁnition,
¤7 § 7 © ¤ 3 It follows that
¤£ ¨ © ¤7 § © & 7 © ¤7 @ ¡ ¢ and
¤ ¥£ ¡ ¦ § ¨ ¤£ Now suppose . Since is a sum of nonnegative terms, it can only be zero if each term is zero, in which case for all and by deﬁnition .
¡ ¢ ¦ § ¤ © § 7© § ¤7 $ ¨ ¦& – p. 5/ Proof that 4 and
§ © satisfy condition 2
, and
$ 1 § 9 1 7© $ 1 9 ¦ 1 & § © 7 7 ¦" & @ © ¦ 7 & § © 7 " " ¡ ¢ ¦ © & & 87¤ 3 Suppose . Then by deﬁnition,
¤7 § 7 © ¤ 3 It follows that
¤£ ¨ © ¤7 § © & 7 © ¤7 @ ¡ ¢ and
¤ ¥£ ¡ ¦ § ¨ ¤£ Now suppose . Since is a sum of nonnegative terms, it can only be zero if each term is zero, in which case for all and by deﬁnition .
¡ ¢ ¦ § ¤ © § ¨ ¦& By a similar argument, is the maximum of a set of nonnegative terms and can only be zero if each term is zero, which again implies that .
¡ ¤ © § © & 7© § ¤7 $ – p. 5/ Proof that 4 and satisfy condition 3 3 By the properties of absolute value,
¤7 ¤7 £ ¨ © 7 § $ 1 9 1 ¤7 ¦ § § 7 © 7 so interchanging 3 and leaves both and unchanged. ¤ § ¡ ¢ ¡ – p. 6/ Proof that
Let
¤ ¦ ¦§ 2 5 3 4 ¡ ¦ § 7 © @ ¢ ¤ ¥£ ¨ 6 . Then
¤7 7 § 7 © @ ¤7 7 2 7 2 7 § satisﬁes condition 4 – p. 7/ Proof that
Let
¤ ¦ ¦§ 2 5 3 4 ¡ ¦ § 7 7 1 @ 7 ¤ £ 7 2 7 2 @ 7 § ¨ @ © ¢ ¤ ¥£ ¨ 6 . Then
¤7 7 § 7 © ¤7 7 2 7 2 7 § satisﬁes condition 4 – p. 7/ Proof that
Let
¤ ¦ ¦§ 2 5 3 4 7 © @ ¤7 7 2 7 2 7 § © ¡ ¦ 2 ¨ ¡ ¦2 § ¨ ¢ £ ¢ ¤£ 7 § ¨ 7 2 7 2 @ @ 7 1 7 ¤ £ @ 7 @ 7 ¡ ¦ § © ¢ ¤ ¥£ ¨ 6 . Then
¤7 7 § 7 © ¤7 7 2 7 2 7 § satisﬁes condition 4 – p. 7/ Proof that
Let
¤ ¦§ ¦ satisﬁes condition 4
. Then for some ,
A ¤£ ¨ © 8B¤ § § B ¦ 3 ,
B 2 B § 6 $ 1 A 1 ¤B 1 B 2 5 ¡ 2 – p. 8/ Let But Proof that and
¤ ¦ ¦ 2 5 ¡ ¤£ ¦ § CB¤ B 2 B § 7 7 2 7 § 7 2 1 7 ¤7 B 2 B 2 B § 1 B 2 1 ¨ © 8B¤ B § ¤B 1 6 3 satisﬁes condition 4
. Then for some ,
A $ 1 A , – p. 8/ ¤B so
¤ ¦ ¦§ 2 5 ¡ ¤£ ¦ § CB¤ B 2 B § 7 1 7 B 2 1 ¨ © 8B¤ B § ¤B 1 6 Let But and Proof that B 2 B 2 B § 1 7 ¤7 7 2 7 7 2 7 § © ¡ ¤£ ¦ 2 ¨ ¡ ¦2 § ¨
– p. 8/ 3 satisﬁes condition 4
. Then for some ,
A 7 2 7 § 7 2 ¤7 B 2 B 2 B § 1 $ 1 A , £ ...
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 Real Numbers, Negative and nonnegative numbers, $ 1 9, $ 1 9 1, 1 9 1 $

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