Uniform Convergence of sequence of derivatives The Question: Let ${(f_{n}): \mathbb{R} \rightarrow \mathbb{R}}$ be a sequence of continuously differentiable functions such that the sequence of derivatives ${f'_{n}: \mathbb{R} \rightarrow \mathbb{R} }$ is uniformily convergent and the sequence ${f_n(0)}$ is convergent. Prove that $f_{n}(x)$ is pointwise convergent. 
The attempt: 
Here is what I have. I need to show $\lim_{n \rightarrow \infty} f_{n}(x) = f(x)$ , or equivently, $\lim_{n \rightarrow \infty} f_{n}(x) -f(x) = 0$
Let $\epsilon > 0$ and let $x_{0} \in \mathbb{R}.$
Then, 
$\lim_{n \rightarrow \infty} (f_{n}(x) -f(x)) = \lim_{n \rightarrow \infty} (x-x_{0})\frac{(f_{n}(x) -f(x))}{(x-x_{0})} = \lim_{n \rightarrow \infty} (x-x_{0})\frac{(f_{n}(x) -f_{n}(x_{0}) +f_{n}(x_{0}) -f(x))}{(x-x_{0})}$. 
This is where I am stuck. I am not sure where to go at this point. Am I on the right track?
Thank you very much!!
 A: Notice that 
$$f_n(x) = f_n(0) + \int_0^x f'_n(t) dt $$
Now let's prove that $(f_n(x))_n$ is a Cauchy sequence :
$$ \left| f_n(x) - f_m (x) \right| \leq  \left| f_n(0) - f_m (0) \right|  + \left|  \int_0^x f'_n(t) - f'_m(t) dt \right| $$
$$\leq  \left| f_n(0) - f_m (0) \right|  +  \int_0^x \| f'_n- f'_m \| dt  $$
$$\leq  \left| f_n(0) - f_m (0) \right|  +   |x| \| f'_n- f'_m \| $$
$$\leq  \left| f_n(0) - f_m (0) \right|  +   |x| \left( \| f'_n- f'\| + \| f'_m - f' \| \right) $$
Now, as $f_n(0)$ is convergent, it a Cauchy sequence, so for $m$ and $n$ big enough, 
$\left| f_n(0) - f_m (0) \right| \leq \epsilon$
And for $m$ and $n$ big enough, you also have $ |x| \left( \| f'_n- f'\| + \| f'_m - f' \| \right) \leq \epsilon$ 
Hence for $m$ and $n$ big enough, $ \left| f_n(x) - f_m (x) \right| \leq 2\epsilon$ : it is a Cauchy sequence, so it converge
A: I think you should try to reformulate your problem with what you know.
($f_n$) is continously differentiable, one can write:
$$f_n (x)=\int_{0}^{x} f'_{n}(t) dt +f_n(0)$$
Likewise, ($f'_n$) is uniformly convergent toward a function that i call f' (continuous thanks to uniform convergence) ; let:
$$f(x)=\int_{0}^{x}f'(t)dt+l$$
with $l=lim (f_n(0) )$
Now you want to consider :
$$f_n(x)-f(x) = (f_n(0)-l) + \int_{0}^{x}[f'_n(t)-f'(t)]dt$$
And you need to prove that both right side terms converge, to zero.
The first term is obvious by definition, the integrale is all that's left :) ... and you might want to consider the uniform convergence here
A: HINTS: 


*

*Use $\epsilon{-}N$ Cauchy characterization of convergent sequences. 

*After write an upper bound of $|f_n(x)-f_m(x)|$ using the convergency of $(f_n(0))$. 

*Last step: use the mean value theorem over the function $f_n-f_m$.
