Yes, that is how to start. You have a Cauchy sequence of functions , and you want to show that the sequence converges to a limit function. The first step is to construct the limit function, the second step is to show that the sequence converges to that limit.

To construct the limit function, notice that for each the sequence of scalars is Cauchy and therefore converges because the scalar field is complete. Call this limit . That defines your limit function . You now have to show that

For this, you need to think of a sequence of functions that converges in the -norm to a non-differentiable function. You haven't said what the space is. If it was something like the interval [–1,1] of the real line, you might look for a sequence of polynomials that converges to |x|, for example.