I am unable to solve this question any hint will be helpful [closed]

Consider the vector space $V:=\{f\,\colon \mathbb{R} \to \mathbb{R}\, \colon f \text{ is a polynomial}\}$.

If$$W =\{f \in V \colon f(m+1) + f(m-1)-2f'(m)=0 \text{ for every integer }\, m, |m|\leq 100\}$$then $\dim(W) <\infty$. True or False with proper justification.