Showing $f$ is a polynomial given sequence Suppose $\{p_k \}_{k=1}^{\infty}$ is a sequence of polynomials and that $f: \mathbb{C} \to \mathbb{C}$ is an entire function such that $p_k(z) \to f(z)$ as $k \to \infty$ with the convergence uniform on each closed disk $D_R=\{z \in \mathbb{C}: \left |z \right |\le R \}$.

A) Show that if there is a finite constant $M$ such that degree$(p_k) \le M$ for each $k$, then $f$ must be a polynomial. 

B) Give an example to show that if there is no such constant $M$,then $f$ might not be a polynomial. 

My approach for $A$,

Suppose $k$=n where $n&lt;+ \infty$ , then $$ \left |f^{(n)}(z) \right |\le \frac{n!}{R^n}M $$

This implies $f^{(n)}$ is constant by Lioville's theorem and therefore $$f^{(n)}(z)=a$$ where $a$ is some finite constant. Therefore integrating both sides we can conclude we get a polynomial of degree $n$.

For part B, I have no clue.
For b, take partial sums for the Maclaurin series of any entire (non-polynomial) function.

For example,
$$
p_n(z) = 1 + z + \frac{z^2}{2!} + \cdots + \frac{z^n}{n!}
$$
converges locally uniformly on $\mathbb{C}$  to $e^z$. (This follows from general theory of convergence for power series. If you don't have that, you can for example use the Weierstrass $M$-test to show uniform convergence on every closed disc.)
For b, take partial sums for the Maclaurin series of any entire (non-polynomial) function.

For example,
$$
p_n(z) = 1 + z + \frac{z^2}{2!} + \cdots + \frac{z^n}{n!}
$$
converges locally uniformly on $\mathbb{C}$  to $e^z$. (This follows from general theory of convergence for power series. If you don't have that, you can for example use the Weierstrass $M$-test to show uniform convergence on every closed disc.)