= Solution
Let the common bound be $M$. The derivative bound and the mean-value theorem give
$$
|f_n(x)-f_n(y)|\leq M|x-y|,
$$
so on every compact interval the sequence is uniformly bounded and equicontinuous. Apply the <Arzelà-Ascoli theorem> first on $[-1,1]$, then to a subsequence on $[-2,2]$, and so on. The <diagonal subsequence for locally uniform convergence> gives a strictly increasing $\phi$ such that $f_{\phi(n)}$ converges uniformly on every $[-R,R]$ to a function $f$. The limit is continuous and satisfies $|f(x)|\leq M$.
Global uniform convergence need not follow. Choose a nonzero continuously differentiable bump function $\psi$ supported in $[-1,1]$ and set
$$
f_n(x)=\psi(x-n).
$$
The functions and their derivatives have a common bound. On every fixed compact interval, $f_n$ is eventually zero, so every subsequence converges locally uniformly to $f=0$. Nevertheless
$$
\sup_{x\in\mathbb R}|f_n(x)-f(x)|
=\|\psi\|_\infty>0
$$
for every $n$. Thus the proposed global conclusion is false.
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