Page 333 - Advanced Linear Algebra

Metric Spaces 317

lim ² ² ³Á ² ³³  B (12.4 )
¦B
(That is, the limit exists and is finite.³
Equivalence Classes of Cauchy Sequences in 4
We would like to define a metric on the set CS ² 4 ³ by
Z

Z
² Á ³ ~ lim ² ² ³Á ² ³³
¦B
However, it is possible that
lim ² ² ³Á ² ³³ ~
¦B
for distinct sequences and , so this does not define a metric. Thus, we are led

to define an equivalence relation on CS²4³ by
— ¯ lim ² ² ³Á ² ³³ ~
¦B
Let CS²4³ be the set of all equivalence classes of Cauchy sequences and
define, for Á  CS ²4 , ³
Z
² Á ³ ~ lim ² ² ³Á ² ³³ (12.5 )
¦B
where  and  .

To see that Z is well-defined, suppose that Z  and Z  . Then since
Z
Z
— and — , we have
Z
Z
( Z Z ( ² ² ³Á ² ³³ c ² ² ³Á ² ³³  ² ² ³Á ² ³³ b ² ² ³Á ² ³³ ¦
as ¦B . Thus,
Z
Z
Z
Z
— and — ¬ lim ² ² ³Á ² ³³ ~ lim ² ² ³Á ² ³³
¦B ¦B
Z
Z
Z
Z
¬ ² Á ³ ~ ² Á ³
Z
Z
which shows that is well-defined. To see that is a metric, we verify the

triangle inequality, leaving the rest to the reader. If Á and are Cauchy

sequences, then
² ² ³Á ² ³³  ² ² ³Á ² ³³ b ² ² ³Á ² ³³
Taking limits gives
lim ² ² ³Á ² ³³  lim ² ² ³Á ² ³³ b lim ² ² ³Á ² ³³
¦B ¦B ¦B