How to prove that this space is complete

Question

I have in $E=\mathcal{C}([a,b])$ (the set of continuous function) this set $M_k$ the set of the $k-$Lipschitzienne function that is $$|f(x)-f(y)|\leq k |x-y|,~ \forall x,y\in [a,b]$$

I want to prove that $(M_k,d_{\infty})$ is a complete space.

Let $(f_n)$ a Cauchy sequence in $(M_k,d_{\infty})$ that is $$\lim_{p,q\in\infty}d_{\infty}(f_p,f_q)=\sup_{x\in[a,b]}|f_p(x)-f_q(x)|=0$$ then we can say that $(f_n(x))$ is Cauchy in the complete metric space $(\mathbb{R},|.|)$ then it converge to $f(x)$

How to prove that $f$ is continuous?

Edit: I don't know how to prove that $(E,d_{\infty})$ is complete.

Thank you

Answer 1

Since $C([a,b])$ is complete it suffices to show that $M_k$ is closed. So, let $f_n \in M_k$ and $f_n \to f$ uniformly. We want to show that $f \in M_k$. This follows from a $3-\epsilon$ argument:

$$|f(x)-f(y)| \le |f(x) - f_n(x)| + |f_n(x) -f_n(y)| +|f_n(y) - f(y)|$$ $$\le 2\|f_n-f\|_\infty + k|x-y| \to k|x-y|.$$