Inverse/Implicit Function Theorems
Hey guys, I need to solve the following problem:
Let B=B(0,r) be an open ball of radius r centered at the origin in R^n. Suppose U is an open subset of R^n containing the closed ball of radius r centered at the origin, f is a function from U to R^n that is differentiable, f(0) = 0, and ||Df(x) - I|| <= s < 1 for all x in the open ball. Prove that if ||y|| < r(1-s), then there is an x in the open ball such that f(x) = y.
I'm pretty sure that this problem uses the inverse and implicit function theorems, but I'm not sure how to start it, and I don't really have any idea what to do with the I that is being subtracted from the derivative matrix. Can somebody please get me pointed in the right direction?