= Solution
{c}
Assume first that the original nonnegative <series> converges to $S$. Its <partial sums> $s_m$ increase to $S$. Write $t_N=\sum_{n=1}^{N}a_{g(n)}$ for the <partial sums> of the <rearrangement of a series>. Since $g$ is an <injection>, the indices $g(1),\ldots,g(N)$ are distinct, so
$$
0\leq t_N\leq s_{\max\{g(1),\ldots,g(N)\}}\leq S.
$$
Thus $(t_N)$ is an increasing <bounded sequence>, and the <bounded monotone sequence theorem> gives $t_N\to T\leq S$.
For any fixed $m$, <surjectivity> of $g$ gives
$$
N_m=\max\{g^{-1}(1),\ldots,g^{-1}(m)\}.
$$
For $N\geq N_m$, nonnegativity implies $t_N\geq s_m$. Taking <limits of sequences> first in $N$ and then in $m$ gives $T\geq S$. Hence $T=S$.
Conversely, $g^{-1}$ is a <bijection>, and the original <series> is the <rearrangement of a series> of $\sum b_n$ defined by $g^{-1}$. The same argument proves its <convergence> whenever $\sum b_n$ converges. \b[Both <series> converge together, and their sums are equal.]
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