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.