For $\mathbb{D}=\{z \in \mathbb{C}: |z| < 1 \},$ it is well known that every function analytic in $\mathbb{D}$ and continuous on the closure of $\mathbb{D}$ can be uniformly approximated by a sequence of rational (in fact polynomials) functions whose poles lies outside the closure of $\mathbb{D}.$ One easy proof is to use the uniform continuity and the Taylor series representation.
I know that the similar statement holds for functions analytic in an annulus $\mathbb{A}=\{z \in \mathbb{C}: r<|z|<R \}$ and continuous on the closure of $\mathbb{A} \ i.e.$ any function which is analytic in $\mathbb{A}$ and continuous on the closure of $\mathbb{A}$ can be uniformly approximated by a sequence of rational functions whose poles lies outside the closure of $\mathbb{A}.$
Is there any simpler proof of the statement that does not use results like Mergelyan’s theorem or Bishop localization lemma? I tried using Runge’s theorem but since the analyticity is given on only in the interior of $\mathbb{A},$ one can’t use the Runge’s theorem.
Any suggestions?