Understanding Proposition 8.7: Operator Norm and Sequences

In summary, Peter found that the final step of the argument is something like this. Suppose we have a number and we let be arbitrary. What can we say about if for all We can actually claim that For, if then suppose Then but contrary to the assumption.
  • #1
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I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ...

I am currently reading Chapter 8: Differentiable Maps and am specifically focused on Section 8.1 Linear Algebra ...

I need some further help in fully understanding the proof of Proposition 8.7 ...Proposition 8.7 and its proof reads as follows:
View attachment 9397
View attachment 9398In the above proof by Browder we read the following:"... ... it follows from Proposition 8.6 that for some . In particular, taking above, we find for every , and hence ... ...

... ... ... "
My question is as follows:Can someone please explain exactly why/how that for every ... implies that ... ... ?In other words if some relation is true for every term of a sequence ... why then is it true for the limit of a sequence ... ...

Help will be much appreciated ...

Peter
 

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  • #2
Let Since is Cauchy in a complete space, it converges to in the space. That is, there exists such that for every we have Now then, we have that

Now we are nearly there.

The final step of the argument is something like this. Suppose we have a number and we let be arbitrary. What can we say about if for all We can actually claim that For, if then suppose Then but contrary to the assumption.

Similarly, for the above argument, because is arbitrary, we can conclude that
 
  • #3
Ackbach said:
Let Since is Cauchy in a complete space, it converges to in the space. That is, there exists such that for every we have Now then, we have that

Now we are nearly there.

The final step of the argument is something like this. Suppose we have a number and we let be arbitrary. What can we say about if for all We can actually claim that For, if then suppose Then but contrary to the assumption.

Similarly, for the above argument, because is arbitrary, we can conclude that

Thanks Ackbach ...

Your post was most helpful ...

Peter
 
  • #4
This is a particular case of the general rule that weak inequalities are preserved by limits (but strict inequalities may not be). If is a sequence with , and for all , then . (But if for all then it need not be true that . All you can assert is that .)
 
  • #5
Opalg said:
This is a particular case of the general rule that weak inequalities are preserved by limits (but strict inequalities may not be). If is a sequence with , and for all , then . (But if for all then it need not be true that . All you can assert is that .)
Thanks for a most helpful post, Opalg ... Peter
 
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