Does there exist a sequence of non-constant polynomials $\big \{p_{n} \big \}_{n \geq 1}$, each $p_n$ in $\mathbb{Z}[X]$, such that $$ \lim_{n \to \infty} p_n =c $$ pointwise on $[0,1]$, where $c$ is a constant?
We can find such polynomials easily on the open interval $(0,1)$, such as $p_n =x^n$, however the closed interval does not seem as simple. The answer seems to be no, but I wouldn't know where to start.
Yes. Consider $p_n(x) = x^{2n} - x^n + 1$. Note that all we need to do is find a sequence of polynomials such that $p_n (1) = p_n (0)$ for all sufficiently large $n$, and with the degrees of all monomial terms going off to infinity.