So for any $\epsilon > 0$ one can find a polynomial $p(z) = \sum_{j=1}^n a_j z^j$ such that $\lvert f(z) - p(z) \rvert < \epsilon$ for all $z \in K.$

Do note that this theorem is not a weaker version of Runge's theorem. Here, we do not need $f$ to be holomorphic on a neighbourhood of $K,$ but just on the interior of $K.$ For example, if the interior of $K$ is empty, then $f$ just needs to be continuous on $K.$ Further, it could be that the closure of the interior of $K$ might not be all of $K.$ Consider $K = D \cup [-10,10],$ where $D$ is the closed unit disc. Then $K$ has two lines coming out of either end of the disc and $f$ needs to only be continuous there.

Also note that this theorem is distinct from the Stone-Weierstrass theorem. The point here is that the polynomials are holomorphic in Mergelyan's theorem.

Bibliography

1

John B. Conway. Functions of One Complex Variable I. Springer-Verlag, New York, New York, 1978.

2

Walter Rudin. Real and Complex Analysis. McGraw-Hill, Boston, Massachusetts, 1987.