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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 8.2 Differentials ... ...

I need some further help in fully understanding the proof of Proposition 8.13 ...

Proposition 8.13 reads as follows:

View attachment 9404

View attachment 9405

I think that a fully detailed proof of Proposition 8.13 reads somewhat as follows:Browder's Definition 8.9 essentially means that \(\displaystyle \text{df}_p\) exists if \(\displaystyle \lim_{ h \to 0 } \frac{1}{| h | } (f(p + h) - f(p) - \text{df}_p h ) = 0\)

Thus ... if we take \(\displaystyle \epsilon = C - \| df_p \|\) then we can find \(\displaystyle \delta\) such that ...\(\displaystyle | | h | \lt \delta \Longrightarrow \frac{1}{ | h | } | (f(p + h) - f(p) - \text{df}_p h ) - 0 | \leq \epsilon\) ... ... ... (1)so that \(\displaystyle | h | \lt \delta \Longrightarrow | (f(p + h) - f(p) - \text{df}_p h ) | \leq \epsilon | h | \) ... ... ... (2)Now the reverse triangle inequality (Duistermaat & Kolk Lemma 1.1.7 (iv) ) states that\(\displaystyle \| x - y \| \geq | \ \| x \| - \| y \| \ |\)Using the reverse triangle inequality we have \(\displaystyle | (f(p + h) - f(p) ) - ( \text{df}_p h ) | \geq | \ | f(p + h) - f(p) | - | \text{df}_p h ) | \ | \) ... ... ... (3)Now (2) and (3) \(\displaystyle \Longrightarrow \)\(\displaystyle | \ | f(p + h) - f(p) | - | \text{df}_p h ) | \ | \leq \epsilon |h|\) \(\displaystyle \Longrightarrow | f(p + h) - f(p) | \leq | \text{df}_p h | + \epsilon |h|\)

Now \(\displaystyle | \text{df}_p h ) | \leq | \text{df}_p | | h ) |\) (Is that correct? ) ... so that ...\(\displaystyle | f(p + h) - f(p) | \leq | \text{df}_p | | h ) | + \epsilon |h|\)

\(\displaystyle \Longrightarrow | f(p + h) - f(p) | \leq ( \| \text{df}_p \| + \epsilon ) |h| + C |h|\)

Is that correct?Now ... my specific problem is how to rigorously and validly make the move \(\displaystyle | \ | f(p + h) - f(p) | - | \text{df}_p h ) | \ | \leq \epsilon |h|\) \(\displaystyle \Longrightarrow | f(p + h) - f(p) | \leq | \text{df}_p h | + \epsilon |h|\)... ... since I have effectively ignored the modulus signs around \(\displaystyle \ | f(p + h) - f(p) | - | \text{df}_p h ) |\) ...

... that is I have assumed that \(\displaystyle | f(p + h) - f(p) | \geq | \text{df}_p h ) |\) ...

Can someone please explain how i deal with this issue ...

Help will be much appreciated ...

Peter

I am currently reading Chapter 8: Differentiable Maps and am specifically focused on Section 8.2 Differentials ... ...

I need some further help in fully understanding the proof of Proposition 8.13 ...

Proposition 8.13 reads as follows:

View attachment 9404

View attachment 9405

I think that a fully detailed proof of Proposition 8.13 reads somewhat as follows:Browder's Definition 8.9 essentially means that \(\displaystyle \text{df}_p\) exists if \(\displaystyle \lim_{ h \to 0 } \frac{1}{| h | } (f(p + h) - f(p) - \text{df}_p h ) = 0\)

Thus ... if we take \(\displaystyle \epsilon = C - \| df_p \|\) then we can find \(\displaystyle \delta\) such that ...\(\displaystyle | | h | \lt \delta \Longrightarrow \frac{1}{ | h | } | (f(p + h) - f(p) - \text{df}_p h ) - 0 | \leq \epsilon\) ... ... ... (1)so that \(\displaystyle | h | \lt \delta \Longrightarrow | (f(p + h) - f(p) - \text{df}_p h ) | \leq \epsilon | h | \) ... ... ... (2)Now the reverse triangle inequality (Duistermaat & Kolk Lemma 1.1.7 (iv) ) states that\(\displaystyle \| x - y \| \geq | \ \| x \| - \| y \| \ |\)Using the reverse triangle inequality we have \(\displaystyle | (f(p + h) - f(p) ) - ( \text{df}_p h ) | \geq | \ | f(p + h) - f(p) | - | \text{df}_p h ) | \ | \) ... ... ... (3)Now (2) and (3) \(\displaystyle \Longrightarrow \)\(\displaystyle | \ | f(p + h) - f(p) | - | \text{df}_p h ) | \ | \leq \epsilon |h|\) \(\displaystyle \Longrightarrow | f(p + h) - f(p) | \leq | \text{df}_p h | + \epsilon |h|\)

Now \(\displaystyle | \text{df}_p h ) | \leq | \text{df}_p | | h ) |\) (Is that correct? ) ... so that ...\(\displaystyle | f(p + h) - f(p) | \leq | \text{df}_p | | h ) | + \epsilon |h|\)

\(\displaystyle \Longrightarrow | f(p + h) - f(p) | \leq ( \| \text{df}_p \| + \epsilon ) |h| + C |h|\)

Is that correct?Now ... my specific problem is how to rigorously and validly make the move \(\displaystyle | \ | f(p + h) - f(p) | - | \text{df}_p h ) | \ | \leq \epsilon |h|\) \(\displaystyle \Longrightarrow | f(p + h) - f(p) | \leq | \text{df}_p h | + \epsilon |h|\)... ... since I have effectively ignored the modulus signs around \(\displaystyle \ | f(p + h) - f(p) | - | \text{df}_p h ) |\) ...

... that is I have assumed that \(\displaystyle | f(p + h) - f(p) | \geq | \text{df}_p h ) |\) ...

Can someone please explain how i deal with this issue ...

Help will be much appreciated ...

Peter

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