Could you clarify in which sense the sequence (f_n) converge to f?
let $(f _n)_n \subset C^1((0,1), \Re)$ be a convergent sequence $ f (0,1) \rightarrow \Re $
assume that $(f _n)_n \subset C^1 ((0,1), \Re) $ converges simply to $g=1$.
Do we have $f \in C^1\left((0,1), \Re \right)$ ?