## Find the closure of a set in C([a,b])

Let $F =\{ f\in C^{1}([a,b]) | \parallel f\parallel \leq M ,\parallel f^{\prime}\parallel \leq N\}$ $M,N>0$ .( $||||$ is the sup norm).
How to show the cloure of $F$ in $(C([a,b]),\parallel \parallel )$ is $A=\{ f\in C([a,b]) | \parallel f\parallel \leq M ,sup_{x\neq y}\frac{|f(x)-f(y)|}{|x-y|}\leq N\}$?
Given $f\in A$,I dont konw how to find an approximate sequence in $F$.