Prove that the limit of a sequence of polynomials a polynomial

Question

Suppose that $(f_n)$ is a convergent sequence of polynomials on D. prove that the limit function f is also a polynomial; or find a counterexample.

Is this equivalent to "the limit of a sequence of constant functions is also constant" ?

Intuitively, I do not think is true because the limit of each polynomial in the sequence is not a polynomial. Is that logic correct ?

there are duplicates to this question but those others have the sequence converging uniformly.

Answer 1

For each $n$, $f_n(z)=\sum_{k=0}^n z^k/k!$ is a polynomial, but $\lim_{n\to\infty}f_n(z)=\exp(z)$ isn't. In addition the convergence is uniform in any bounded region of the complex plane.