Do you know any theorem that will help me with this question:
Let $f$ be any continuous function on complex plane. Show that there is a sequence $(P_n)$ of polynomials such that $P_n$ converges pointwise to $f$ ($P_n(z) \to f(z)$ for all $z$).
I tried to use the Runge's approximation theorem, but I have an arbitrary continuous function, and the theorem requires analyticity on the interior.
I take the rational functions to be with poles only at infinity so they will be polynomials, but I am struggling with the function being only continuous.
In fact, if $f_n$ is a sequence of holomorphic functions converging pointwise to a function $f$ on some domain $\Omega$, a (not as well-known as it should be) theorem by Osgood shows that there must exist an open, dense subset $U \subseteq \Omega$ such that $f$ is holomorphic on $U$ (and that the convergence is in fact locally uniform on $U$).
In particular this for example implies that a nowhere holomorphic function cannot be approximated pointwise by polynomials on any domain.