= Solution
<Runge theorem> says that if $K\subset\mathbb C$ is compact and $\mathbb C\setminus K$ is connected, every <function> holomorphic near $K$ is uniformly approximable on $K$ by <polynomials>.
Choose a compact exhaustion $K_j$ of the slit disc $\Delta\setminus L^+$, rounding the two sides of the slit so that $0\notin K_j$ and $\mathbb C\setminus K_j$ is connected. Runge applied to $1/z$ gives a <polynomial> $p_j$ with $\sup_{K_j}|p_j-1/z|<1/j$. Every compact subset of the slit disc lies in all sufficiently large $K_j$, proving locally <uniform convergence>.
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