Page 225 - DMTH503_TOPOLOGY

Unit 27: Complete Metric Spaces

Show that Notes
(i) <b > is a Cauchy sequence.
n
(ii) <a > converges to a point pX iff <b > converges in p.
n n
Solution: Let <a > be a Cauchy sequence in a metric space (X,) so that
n
given , K > 0 n N s.t.
o
n, mn  (a , a ) <K …(1)
o n m
Also let <b > be a sequence in X s.t.
n
1
(a , b ) < nN …(2)
n n
n
Step (i): To prove that <b > is a Cauchy sequence.
n
Let, K > 0 any given real numbers.
1
Then m N s.t. <k. …(3)
o
m
o
Set K = max. (n , m ).
o o o
Then K n , m , so that
o o o
1 1 1
 , …(4)
K o n o m o
1 1 1 1 1 1
< K,    < K <K. …(5)
m K m K m K
o o o o o o
1 1
If n  K , then (a , b ) <  < K,
o n n
n K
o
i.e., (a , b ) < K  n, m  K …(6)
n n o
For n, m  K , we have
o
(b , b )  (b , a ) + p(a , a ) + p(a , b )
n m n n n m m m
< K + K + K = 3 K.

1
Choosing initially K = , we get
3
(b , b ) <  n  K .
n m o
This proves that <b > is a Cauchy sequence.
n
Step (ii): Let a ® p  X.
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To prove that b ® p.
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a ® p  given , K > 0,  m  N s.t.
n o
n  m  (a , p) < K.
o n
We have seen that <a > and <b > are Cauchy Sequences and therefore given , K > 0,  n  N s.t.
n n o
 m, n  n  (a , a ) < K, (b , b ) < K.
o n m n m

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