= Solution
Suppose for a contradiction that $b_0\ne0$. The <Eisenstein series> in the question has constant term $2\zeta(k)$, so
$$
h=g-\frac{b_0}{2\zeta(k)}G_k
$$
has zero constant term and is therefore a level-one <cusp form> of weight $k$. Part (b) gives the coefficient bound $[q^n]h=O(n^{k/2})$. Since $b_n=O(n^{k/2})$ as well, the displayed Fourier expansion of $G_k$ would imply
$$
\sigma_{k-1}(n)=O(n^{k/2}).
$$
Take $n=\ell$ through the <prime number>[primes]. Then
$$
\sigma_{k-1}(\ell)=1+\ell^{k-1},
$$
which cannot be $O(\ell^{k/2})$ because $k-1>k/2$ for $k\geq4$. Therefore $b_0=0$, so $g$ vanishes at the only cusp of $\Gamma(1)$ and belongs to $S_k(\Gamma(1))$. This is the <Fourier coefficient growth criterion for a level-one cusp form>.
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