= Solution
For each $0<r<1$, the spherical <mean value property for harmonic functions> gives
$$
\int_{S^{d-1}}u(x+\varepsilon r\omega)\,d\omega=d\omega_d u(x).
$$
Insert this into part 1(iii). The normalization of the radial <mollifier> is $d\omega_d\int_0^1\phi(r)r^{d-1}dr=1$, hence
$$
\boxed{(u*\phi_\varepsilon)(x)=u(x).}
$$
A convolution with a <smooth function> of compact support is smooth wherever it is defined. Every point admits such an $\varepsilon$, so $u\in C^\infty(\Omega)$.
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