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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 $$S_m \to S$$ for some $$S \in \mathscr{L} ( \mathbb{R}^n)$$. In particular, taking $$m = 0$$ above, we find $$\| I - S_p \| \leq \frac{t}{ 1 - t }$$for every $$p$$, and hence $$\| I - S \| \leq t/(1 - t )$$ ... ...
... ... ... "
My question is as follows:Can someone please explain exactly why/how that $$\| I - S_p \| \leq \frac{t}{ 1 - t }$$for every $$p$$ ... implies that $$\| I - S \| \leq t/(1 - t )$$ ... ... ?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
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 $$S_m \to S$$ for some $$S \in \mathscr{L} ( \mathbb{R}^n)$$. In particular, taking $$m = 0$$ above, we find $$\| I - S_p \| \leq \frac{t}{ 1 - t }$$for every $$p$$, and hence $$\| I - S \| \leq t/(1 - t )$$ ... ...
... ... ... "
My question is as follows:Can someone please explain exactly why/how that $$\| I - S_p \| \leq \frac{t}{ 1 - t }$$for every $$p$$ ... implies that $$\| I - S \| \leq t/(1 - t )$$ ... ... ?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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