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