I'm looking at Lemma 2.1 in Reddien's paper "On Newton's Method for Singular Problems". 
I'm still fairly new to Banach spaces. So I'd appreciate a hint as to why Lemma 2.1 is an "easy" consequence of Taylor's theorem. I'm confused on both part (i) and (ii), but I'm hoping some help with part (i) will allow me to understand part (ii) on my own.
Edit: Notation: $N_1$ is the null space of $F'(x^*)$, and $X_1$ is closed subspace such that $F'(x^*)X_1=X_1$ and $X=N_1\oplus X_1$. We assume that $F\in C^3$.
My current idea is this: by Taylor's theorem, for $y\in X_1$ we can write $ =F'(x^*)y+F''(x^*)(x-x^*)y+o(||x-x^*||^2)y.$ Then by the triangle inequality $||F'(x)y||≥\bigg(\frac{||F'(x^*)y||}{||y||}−||F''(x^*)||\,||x-x^*||−o(||x-x^*||^2)||\bigg)||y||.$ So now we can pick $x$ close enough to $x^*$ so that the term in front of $||y||$ is positive, and I think we can bound $\frac{||F'(x^*)y||}{||y||}$ away from zero since the range of $F'(x^*)$ is closed, that being $X_1$.