Prove that a map is a homeomorphism and the inverse is bounded

Question

I'm trying to unravel an obscure passage in a textbook, which states that if $\phi :\mathbb{R}^m\to\mathbb{R}^m$ is continuous, bounded and Lipschitz with constant $\varepsilon$ (which is still free to choose) and $A := Df(0)$, where $f$ is smooth map with a hyperbolic fixed point at $0$, then the following statements are true:

1. "It is easy to see the if $\varepsilon$ is small enough then $A+\phi$ is a homeomorphism"
2. $\left\|(A+\phi)^{-1}\right\| \le 1$ with $\|\cdot\|$ the sup-norm.

I don't find it easy to see that the first statement is true, and I also cannot reproduce the second assertion. Does anyone know how to proceed here?