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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 \(\displaystyle S_m \to S\) for some \(\displaystyle S \in \mathscr{L} ( \mathbb{R}^n)\). In particular, taking \(\displaystyle m = 0\) above, we find \(\displaystyle \| I - S_p \| \leq \frac{t}{ 1 - t }\)for every \(\displaystyle p\), and hence \(\displaystyle \| I - S \| \leq t/(1 - t )\) ... ...

... ... ... "

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

... ... ... "

My question is as follows:Can someone please explain exactly why/how that \(\displaystyle \| I - S_p \| \leq \frac{t}{ 1 - t }\)for every \(\displaystyle p\) ... implies that \(\displaystyle \| I - S \| \leq t/(1 - t )\) ... ... ?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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