Contraction mapping problem in n-dimensional vector-valued function

Let be a closed subset of which contains the entire line segment between any two of its points and let be a continuously differentiable map from an open subset of containing into . Suppose that and that there is a real number such that

for all . Prove that the restriction of to is a contraction map, so that the fixed point theorem is applicable.

...

I have my idea, but I could not go through. I need to show that for any and

where

by the Mean Value Theorem for several variables, we have

+

where is the point on the line segment between and which yields

From here, I have no idea to proceed to make the desired inequality. There is a hint said that I could use the preceding problem in the book as follows:

If , its length is

but I could not relate how I can use it. The assumption that should be the hint that and the fact that is closed ensures the existence of cauchy sequences converging to a fixed point.

The Schwarz inequality seems to be the way to go, but I could not utilize it.

so that I could somehow use to relate that

Anyone has an idea on how to proceed ? Thanks.

EDITED: FORGET WHAT I WROTE ABOVE. I was careless. The Mean Value Theorem cannot be applied here. I wrongly wrote which is not correct. In face