Let a sequence $f_k$ where $f_k:\mathbb {R^n} \rightarrow \mathbb{R^n}$ and each $f_k$ is a contraction.
Defining $F_k=f_0 \circ ... \circ f_k$ i want to show that $F_k \rightarrow C$ where $C$ is Constant and the convergence is uniform
Proving that $F_k$ converges to a Constant is easy, but i don't know How to show this convergence is uniform
Edit: I have seen my mistake and accepted the answer, indeed we need the assumption of bounded derivative on each function of sequence otherwise one can play with coefficients
For each nonnegative integer $u$, let $$a_u=1-\frac{1}{(u+2)^2}$$ and let $f_u:\mathbb{R}^n\to\mathbb{R}^n$ be defined by $$f_u(x)=a_u x$$ It's easily seen that each $f_u$ is a contraction.
But for each $x\in\mathbb{R}^n$, we have \begin{align*} \lim_{k\to\infty}F_k(x) &=\lim_{k\to\infty}(f_0\circ\cdots\circ f_k)(x)\\[4pt] &=\left(\lim_{k\to\infty}\left(\prod_{u=0}^k a_u\right)\right)\!x\\[4pt] &=\left(\prod_{u=0}^\infty a_u\right)\!x\\[4pt] &=\left(\prod_{u=0}^\infty \left(1-\frac{1}{(u+2)^2}\right)\right)\!x\\[4pt] &=\left({\small{\frac{1}{2}}}\right)\!x\\[4pt] \end{align*} so $F_k$ does not approach a constant.