I suppose that should be for all $ | f^{(j)}(x)| \leq M $, not $R$.
Differentiability implies continuity, so since the function is infinitely differentiable, every derivative is continuous. Continuous functions map compact sets to compact sets, and in $\mathbb{R}$, by Heine-Borel, a set is compact iff it is closed and bounded.
Thus, the continuous functions $f, f' $ and $f''$ will map $ \overline{ B(0,R)} = \{ x\in \mathbb{R} : |x| \leq R \}$ to closed and bounded sets. Hence there exists positive constants $A,B,C$ such that $$f\left( \overline{ B(0,R)} \right) \subseteq \overline{ B(0,A)} ,$$
$$ f'\left( \overline{ B(0,R)} \right) \subseteq \overline{ B(0,B)} ,$$
$$ f''\left( \overline{ B(0,R)} \right) \subseteq \overline{ B(0,C)}. $$
Thus for all $x\in \overline{ B(0,R)}$, $$ f^{(j)} (x) = f^{ (j\mod 3) } (x) \in \overline{ B(0,\max\{A,B,C\})}$$
as required.