= Solution
The polynomial form of <Runge theorem> says that if $K\subset\mathbb C$ is <compact set>[compact], $\mathbb C\setminus K$ is connected, and $f$ is <holomorphic function>[holomorphic] on a neighbourhood of $K$, then for every $\varepsilon>0$ there is a <polynomial> $P$ such that
$$
\sup_{z\in K}|f(z)-P(z)|<\varepsilon.
$$
Statement (i) is false. Take the <entire function> $f(z)=e^{-z}$. It is bounded on the closed first quadrant $E$. If polynomials $P_n$ converged <uniform convergence>[uniformly] to $f$ on $E$, then each sufficiently late $P_n$ would be bounded on the positive real axis. A polynomial bounded on that ray must be constant. No sequence of constants converges uniformly to $e^{-x}$ on $[0,\infty)$, since its values at zero and near infinity differ by one.
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