Inequality in chapter "pontryagin construction" from Milnor's topology from the differentiable viewpoint

Question

at the moment I am reading Milnor's book "topology from the differentiable viewpoint". I am stuck on page 49 (chapter "the pontryagin construction", proof of lemma 4). There he claims, that for $0<||u||<\min\{1/c_1, 1\}$ we have the inequality \begin{align} F(x,u)\cdot u\geq||u||^2-c_1||u||^3. \end{align} Prior, he stated the two other inqualities

\begin{align} ||F(x,u)-u||\leq c_1||u||^2 \ \ \ \text{for ||u||<1 and} \\ |(F(x,u)-u)\cdot u|\leq c_1||u||^3 \end{align}

I can deduce the inequality in the case that $(F(x,u)-u)\cdot u\leq$ 0 but what about the case when it is gereater than 0?

Answer 1

We have

\begin{equation} \Vert u \Vert^2 - F(x,u) \cdot u \leq |\Vert u \Vert^2 - F(x,u) \cdot u| = |F(x,u) \cdot u - \Vert u \Vert^2| = |(F(x,u) - u) \cdot u| \leq c_1 \Vert u \Vert^3 \end{equation}

Hence

\begin{equation} -F(x,u) \cdot u \leq c_1 \Vert u \Vert^3 - \Vert u \Vert^2 \Leftrightarrow F(x,u) \cdot u \geq \Vert u \Vert^2 - c_1 \Vert u \Vert^3 \end{equation}

The fact that $\Vert u \Vert^2 - c_1 \Vert u \Vert^3 > 0$ follows from $0 < \Vert u \Vert < \min\{c_1^{-1}, 1\}$.