$\lim_{n_{i}\to \infty} [g_{n_{i}}]= [g] \in \mcal{Z}_{d,\leq k}(X_{\C}\times \A^{q}_{\C})$ for its limit. Note that $[g]$ satisfies $[g] +\overline{[g]} = \Gamma(\gamma)$. Since $\Gamma$ is injective and its image consists precisely of equidimensional cycles, we are done if we can find an equidimensional cycle $[g']$ such that $[g'] + \overline{[g']} = [g] + \overline{[g]}$.

Choose a representative $\gamma\in \Mor{\C}{X_{\C}}{\mcal{C}_0(\P^q_{\C})}^{+}$ of $[\gamma]$. Choose a representative $g= \sum n_V V\in \mcal{Z}_{d}(X_{\C}\times\P^{q}_{\C})$ of $[g]$ such that if $n_V \neq 0$ then $V\nsubseteq\P^{q-1}\times X$. Since $[g] + \overline{[g]} = \Gamma([\gamma]) \in 
\mcal{Z}_{d}(X_{\C}\times \A_{\C}^{q})$ we see that $g + \overline{g} = \sum (n_V + 
n_{\overline{V}} )V = \Gamma(\gamma) + h$ where $h\in \mcal{Z}_{d}(X_{\C}\times \P_{\C}^{q-1})$. Write $h 
= \Sigma m_{W} W$. Since $V\nsubseteq \P^{q-1}$ whenever $n_{V}\neq 0$ we see that if $m_{W} \neq 0$ then a term of $-m_{W}W$ must appear in $\Gamma(\gamma)$. In particular $h$ is equidimensional. Consequently $g + \overline{g}$ is equidimensional.

If $n_{V} + n_{\overline{V}} \neq 0$ then $V$ is equidimensional. Define $$g' = \sum_{n_{V} + n_{\overline{V}}\neq 0} n_{V} V.$$ Since $g'$ is an equidimensional cycle there is an $f\in (\Mor{\C}{X_{\C}}{\mcal{C}_0(\P^q_{\C})}^{+})^{G}$ such that $\Gamma(f) = g'$. Since $\Gamma([f] + \overline{[f]}) = [g'] + \overline{[g']} = [g] + \overline{[g]} = \Gamma([c])$ and $\Gamma$ is injective, we conclude 