This theorem requires two hypotheses: the function is one-to-one and its differential is invertible. Here are the first steps for a proof:
1) apply the mean value theorem to to prove that is one-to-one on (hint: the differential of is ...)
2) prove that the differential is not singular. (This is standard. If you've never seen this, justify the following: if is a linear map such that , then the series converges and its sum satisfies so that is invertible)
3) apply the global inverse function theorem.