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Let ${f_n}$ be a sequence of real differentiable functions defined in the interval $[0,1]$ such that there exists a point $x_0$ for which ${f_n(x_0)}$ is convergent. Moreover, the sequence of derivatives ${f'_n}$ is uniformely bounded in the interval $[0,1]$. Then prove that ${f_n}$ has a convergent partial sequence. I have the feeling that I must somehow use Ascoli-Artzelà's theorem at some point. I've tried to prove that the sequence is uniformely bounded using the fact that the sequence of derivatives is bounded, but I only manage to do it in a small open set containing $x_0$. For the equicontinuity something similar happens. Any ideas?

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    So far you have asked 11 questions but have accepted not a single answer. Please think about it. – 2017-02-18
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    sry didn't know that was necessary – 2017-02-18
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    It is not necessary, and if you feel that a question hasn't been given any satisfactory answer, then you need not accept any of them. When a satisfactory answer is given, though, you are strongly encouraged to accept it (you may accept just one answer, but if several good ones are provided, you may upvote as many of them as you please). – 2017-02-18
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    I'll keep that in mind from now on. Thanks! – 2017-02-18

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Hint: Yes, Arzela-Ascoli is the way to go. For equicontinuity: Use the MVT to see $f_n(y) - f_n(x)= f_n'(c)(y-x).$ For uniform boundedness, take $x= x_0.$