 Solution

Let the common bound be $M$. The derivative bound and the mean-value theorem giveso on every compact interval the sequence is uniformly bounded and equicontinuous. Apply the Arzela-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 $\varphi$ such that $f_{\varphi (n)}$ converges uniformly on every $[-R,R]$ to a function $f$. The limit is continuous and satisfies $|f(x)|\le M$.

Global uniform convergence need not follow. Choose a nonzero continuously differentiable bump function $\psi$ supported in $[-1,1]$ and setThe 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 $f=0$. Neverthelessfor every $n$. Thus the proposed global conclusion is false.