I'm reading this book by S. Kobayashi and K. Nomizu, "Foundations of Differential Geometry, Vol.1" and I have a problem in the proof of this lemma at the page 170:
And the proof gose like this:
My problem is that I don't understand why we can assume the last double inequality. My only idea is to look at $g$ in normal coordinates and thus $g_{ij}(x)=\delta_{ij}-\frac{1}{3}R_{iajb}x^ax^b+O(\epsilon^3)$. Can someone help me with some details please?