Showing the following sequence of functions are uniformly convergent 
Let $f_n:[0,1]\to \mathbb{R}$ be a sequence of continuously differentiable  functions such that 
$$f_n(0)=0,\:\: |f_n'(x)|\leq 1, \text{for all }n\geq 1, x\in (0,1).$$ 
  Suppose further that $f_n(.)$ is convergent to some function $f(.)$. Show that $f_n(.)$ converges to $f(.)$ uniformly. 

I tired to prove this problem, but I'm lost and I cannot find a correct approach.
Firstly, I don't know if $f(.)$ is continuous. There must be some way to show $f$ is continuous, but I can't and I'm not sure if I need it. Secondly, I've obtained that $(f_n(.))$ are uniformly bounded and this because of 
$$\text{for some }z\in(0,x)\subset(0,1),\:\:|f'_n(z)|=|\frac{f_n(x)-f_n(0)}{x-0}|\leq 1\to |f_n(x)|< 1.$$
I tried to show $(f_n(.)) $ is equicontinuous. Here is my work.
Let $x\in (0,1),$ and let $\epsilon>0$. As $|f_n'(x)|\leq 1$, we can write $\lim_{y\to x}|\frac{f_n(x)-f_n(y)}{y-x}|\leq 1$, hence $|f_n(x)-f_n(y)|\leq |x-y|$. So if we take $\delta=\epsilon$,then for any $y$ that $|x-y|<\delta$, we have $|f_n(x)-f_n(y)|<\epsilon.$ So $(f_n)$ is equicontinuous on $(0,1)$. 
Also we know $f_n(0)=0$, so this sequence is equicontinuous on $x=0$. If I'm correct in each of these steps, probably I'm done with the proof by showing that (f_n(.)) is equicontinuous on $x=1$. 
Also in my proof, I didn't use this fact that $(f_n())$ are continuously differnetiable. I also saw the following proof, for this quetsion, but honestly, I think, it's not totally correct, since they use the continuity of $f(.)$ without proving it. here is the link "Show that $f_n(\cdot)$ is uniformly convergent."

Let $\varepsilon>0$ be given, and set $\|\, f_k'\| = \sup \left[|\,f_k'(x)|: x \in (0,1) \right]$  $(k=1,2,\ldots)$. Since the collection of open balls $\mathcal{B}: = \{B(\, x, \frac{\varepsilon}{3}) : x \in [0,1] \}$ is a cover for $[0,1]$, we may find a finite subcover, say $B(\,x_1, \frac{\varepsilon}{3}),  \, \ldots, \, B(\,x_M, \frac{\varepsilon}{3})$ (Heine-Borel Theorem).  Since $f_n$ converges pointwise on $[0,1]$, for each point $x_j \: \left(\,j=1,\ldots, M \right)$ we may find a positive integer $N_j$ so that 
\begin{equation} \left|\, f_n(x_j) -
 f_m(x_j) \right| <  \frac{\varepsilon}{3}  \text{ whenever } n, m \geq N_j \,.
\end{equation}
Setting $N = \max [N_1,  \ldots, N_M]$ shows that
\begin{aligned}
\left|\,f_n(x)- f_m(x) \right| & \leq  \left| \,f_n (x)- f_n(x_j) \right| + \left|\, f_n (x_j)- f_m(x_j) \right| + \left|\, f_m(x_j)- f_m(x) \right| \\
& <  \| \,f_n'\||x-x_j| + \frac{\varepsilon}{3} + \|\,f_m'\| |x_j-x| 
\\
& \leq \frac{\varepsilon}{3} + \frac{\varepsilon}{3} + \frac{\varepsilon}{3}= \varepsilon
\; \: \text{ whenever } \, n,m \geq N \text{ and } x \in [0,1] . 
\end{aligned}
Since $\mathbb{R}$ is complete (or rather $f(x):=\lim_{n \to \infty} f_n(x)$ exists for $x \in [0,1]$ ), it follows that the sequence of functions $\{\,f_n\}_{n=1}^\infty$ converges uniformly on $[0,1]$ (Cauchy Criterion).
