Page 335 - Advanced Linear Algebra
P. 335
Metric Spaces 319
We can think of as a constant approximation to , with error at most ° .
Let be the sequence of these constant approximations:
² ³ ~
This is a Cauchy sequence in 4 . Intuitively speaking, since the 's get closer
to each other as ¦ B , so do the constant approximations. In particular, we
have
Z
² Á ³ ~ ²´ µÁ ´ µ³
Z
Z
Z
²´ µÁ ³ b ² Á ³ b ² Á ´ µ³
b Z ² Á ³ b ¦
as Á ¦ B . To see that converges to , observe that
Z
Z
Z
² Á ³ ² Á ´ µ³ b ²´ µÁ ³ b lim ² Á ² ³³
¦B
~ b lim ² Á ³
¦B
Now, since is a Cauchy sequence, for any , there is an such that
5
Á 5 ¬ ² Á ³
In particular,
5 ¬ lim ² Á ³
¦B
and so
Z
5 ¬ ² Á ³ b
which implies that ¦ , as desired.
Uniqueness
ZZ
ZZ
Z
Z
Finally, we must show that if ²4 Á ³ and ²4 Á ³ are both completions of
Z
Z
²4Á ³, then 4 4 . Note that we have bijective isometries
Z
Z
¢4 ¦ 4 4 and ¢4 ¦ 4 4 Z Z
Hence, the map
~ c ¢ 4 ¦ 4
²
is a bijective isometry from onto 4 , where 4 4 is dense in 4 Z . See
Figure 12.4.³
