MHB Proof of Lemma 2.1, Part (vi) in Palka's Ch.4: Explaining Inequality

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I am reading Bruce P. Palka's book: An Introduction to Complex Function Theory ...

I am focused on Chapter 4: Complex Integration, Section 2.2 Properties of Contour Integrals ...

I need some further help with some other aspects of the proof of Lemma 2.1, part (vi), Section 2.2, Chapter 4 ...

Lemma 2.1, Chapter 4 reads as follows:View attachment 7433
View attachment 7434In the above text from Palka, near the end of the proof of part (vi), we read the following:" ... ... $$\int_a^b \text{ Re} \{ u f[ \gamma (t) ] \dot{ \gamma } (t) \} \ dt$$ $$\le \int_a^b \lvert u f[ \gamma (t) ] \dot{ \gamma } (t) \rvert \ dt$$ ... ... "
Can some please explain why/how we have $$\int_a^b \text{ Re} \{ u f[ \gamma (t) ] \dot{ \gamma } (t) \} \ dt $$ $$\le \int_a^b \lvert u f[ \gamma (t) ] \dot{ \gamma } (t) \rvert \ d$$t ... ... ?
Help will be much appreciated ...

Peter
 
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It follows from the fact that $\operatorname{Re}(z) \le \lvert z\rvert$ for all $z$.
 
Euge said:
It follows from the fact that $\operatorname{Re}(z) \le \lvert z\rvert$ for all $z$.
Oh ... should have seen that ...

... thanks Euge ...

Peter
 
A sphere as topological manifold can be defined by gluing together the boundary of two disk. Basically one starts assigning each disk the subspace topology from ##\mathbb R^2## and then taking the quotient topology obtained by gluing their boundaries. Starting from the above definition of 2-sphere as topological manifold, shows that it is homeomorphic to the "embedded" sphere understood as subset of ##\mathbb R^3## in the subspace topology.
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