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I am unable to figure out the proof of Order of convergence of the Fixed point method,

source - http://math.bd.psu.edu/faculty/stevens/Old-Courses/MTHBD423-Fall2003/Notes-Handouts/fixed-point.pdf

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I doubt in page 2 that why = was taken and not $\leq$ ?

Any help guys also if any reference is there where it is lucid kindly suggest !

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Look at point $a)$ second line. Equality is given by the mean value theorem. It is just that.

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    I missed it ! thanks. – 2017-02-26