Page 330 - Advanced Linear Algebra

314 Advanced Linear Algebra

Figure 12.3

We leave it to the reader to show that the sequence ² ²%³³ is Cauchy, but does

(
not converge in ²*´ Á µÁ ³ . The sequence converges to a function that is not

continuous.)
Example 12.13 The metric space M B is complete. To see this, suppose that % ² ³
is a Cauchy sequence in M B , where

% ~ ²% Á Á% Á 2 Áó
Then, for each coordinate position , we have

( c Á (%  Á ( % c Á % (% sup ¦ Á as Á ¦ B ( 12.2)

Hence, for each , the sequence % ² Á ³ of th coordinates is a Cauchy sequence in
s
s ( d or ) . Since ( d or ) is complete, we have
²% ³ ¦ & as ¦ B
Á
for each coordinate position . We want to show that & ~ ² & ³  M B and that
²% ³ ¦ &.

)
²
Letting ¦B in 12.2 gives
sup ( Á c & (% ¦ as ¦ B ( 12.3)

and so, for some ,

( c Á & (%  for all

and so
((& b (% Á ( for all

But since % M B , it is a bounded sequence and therefore so is ²& ³ . That is,

B
)
B
& ~ ²& ³  M . Since ²12.3 implies that ²% ³ ¦ &, we see that M is

complete.