Can we find a sequence of polynomials $\{p_n \}$, such that $p_n(0) \rightarrow 1$ as $n \rightarrow \infty$ and for $z \ne 0$, $p_n(z) \rightarrow 0$ as $n \rightarrow \infty$?
My thought: I think yes, we can find such that sequence. If we can write this polynomial sequence by constructions, then the constant term $p_n(z)$ must be $1$ or approaches to $1$ as $n \rightarrow \infty$, then $p_n(z)-1$ should be a zero function after certain $n \in \mathbb{N}$. Does that argument work? or may be we cannot find such this sequence?