Why is Re(z) ≤ |z| in Palka's Lemma 2.1(vi) proof?

[Math Amateur]

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

 

[Euge]

It follows from the fact that $\operatorname{Re}(z) \le \lvert z\rvert$ for all $z$.

 

[Math Amateur]

It follows from the fact that $\operatorname{Re}(z) \le \lvert z\rvert$ for all $z$.

Oh ... should have seen that ...

... thanks Euge ...

Peter