that $$\begin{aligned}
&&\mathbb{E}\Big[\Big(\sum_{i=1}^{\infty}\sqrt{\lambda_{i}}\|\beta_{i}^{H,N,n}-\beta_{i}^{H,N}\|_{\alpha,0,T}\Big)^2\Big]\cr 
&\leq& C(\frac{1}{n})^{2(\varpi+\alpha-1)}\Big(\sum_{i=1}^{\infty}\sqrt{\lambda_{i}}\Big)^2\rightarrow 0 \qquad ~{\rm as}~ n \rightarrow \infty.\end{aligned}$$

Due to Lemma \[lemboundNm\] and Markov’s inequality, we see that for all $\varepsilon>0, R \geq 1$, $$\mathbb{P}\big(\|u^{N,n}-u^{N,m}\|_{\alpha,T}>\varepsilon, \|u^{N,n}\|_{\alpha,T} \leq R, \|u^{N,m}\|_{\alpha,T} \leq R \big) \rightarrow 0\qquad ~{\rm as}~n \rightarrow \infty,$$ and $$\limsup_{n,m \rightarrow \infty}\mathbb{P}\big( \|u^{N,n}\|_{\alpha,T} > R~{\rm or}~\|u^{N,m}\|_{\alpha,T} > R \big) \leq 2 \sup_{n\in\mathbb{N}} \mathbb{P}\big( \|u^{N,n}\|_{\alpha,T} > R\big).$$ Next, Lemma \[lemboundN\] and Markov’s inequality imply that $$\sup_{n\in\mathbb{N}} \mathbb{P}\big( \|u^{N,n}\|_{\alpha,T} > R\big)\rightarrow 0 \qquad ~{\rm as}~ R \rightarrow \infty   .$$ Thus, $$\|u^{N,n}-u^{N,m}\|_{\alpha,T}\rightarrow0\qquad ~{\rm as}~n,m \rightarrow \infty,$$ in probability. Then there exists a random process $u^N$ such that $$\|u^{N,n}-u^{N}\|_{\alpha,T}\rightarrow0,\qquad ~{\rm as}~ n \rightarrow \infty,$$ in probability. Denoting an almost surely convergent subsequence by the same symbol, we have $$\|u^{N,n}-u^{N}\|_{\alpha,T}\rightarrow 0\qquad ~{\rm as}~n \rightarrow \infty,$$ almost surely.

Since $\|u^{N,n}-u^{N}\|_{\alpha,T}\rightarrow0, ~{\rm as}~n \rightarrow \infty,$ a.s., we easily obtain $$\int_{0}^{t} S_{t-s} f(u^{N,n}_{s})\mathrm{d}s \rightarrow \int_{0}^{t} S_{t-s} f(u^{N}_{s}) \mathrm{d}s\qquad ~{\rm as}~n \rightarrow \infty,$$ almost surely.

Similar to the cases of the proof in Lemma \[lemboundNm\], denoting $\mathbf{1}_{t}:=\mathbf{1}_{\{\|u^{N,n}\|_{\alpha,t}\leq R, \|u^{N}\|_{\alpha,t} \leq R\}}$, we have $$\begin{aligned}
&& \Big|\sum_{i=1}^{\infty}\sqrt{\lambda_{i}}\int_{0}^{t} S_{t-s}  (g(u^{N,n}_{s})-g(u^{N}_{s}))e_i \mathrm{d} \beta_{i}^{H,N,n}(s)\mathbf{1}_{t}\Big|^2\cr
&\leq& C \Big(\sum_{i=1}^{\infty}\sqrt{\lambda_{i}}\|\beta_{i}^{H,N,n}\|_{\alpha,0,T}
\int_{0}^t[r^{-\alpha}+(t-r)^{-\alpha}]|u^{N,n}_r-u^{N}_r|
\mathrm{d}r\mathbf{1}_{t}\Big)^2 \cr
&&+C \Big(\sum_{i=1}^{\infty}\sqrt{\lambda_{i}}\|\beta_{i}^{H,N,n}\|_{\alpha,0,T}
\int_{0}^t|u^{N,n}_r-u^{N}_r|\big(\int_{0}^{r}\frac{|u^{N,n}_r-u^{N,n}_q|}{(r-q)^{1+\alpha}}\mathrm{d}q\big)\mathrm{d}r\mathbf{1}_{t}\Big)^2 
\cr
&&+C \Big(\sum_{i=1}^{\infty}\sqrt{\lambda_{i}}\|\beta_{i}^{H,N,n}\|_{\alpha,0,T}
\int_{0}^t|u^{N,n}_r-u^{N}_r|\Big(\int_{0}^{r}\frac{|u^{N}_r-u^{N}_q|}{(r-q)^{1+\alpha}}\mathrm{d}q\Big)\mathrm{d}r\mathbf{1}_{t}\Big)^2 
\cr
&&+C \Big(\sum_{i=1}^{\infty}\sqrt{\lambda_{i}}\|\beta_{i}^{H,N,n}\|_{\alpha,0,T}
\int_{0}^t\Big(\int_{0}^{r}\frac{|u^{N,n}_r-u^{N}_r-u^{N,n}_q+u^{N}_q|}{(r-q)^{1+\alpha}}\mathrm{d}q\Big)\mathrm{d}r\mathbf{1}_{t}\Big)^2 \cr
&\leq& C_{N,R}
\int_{0}^t\|u^{N,n}-u^{N}\|^2_{\alpha,r}\mathbf{1}_{r}\mathrm{d}r+C_{N} \|u^{N,n}-u^{N}\|^2_{\alpha,t}\mathbf{1}_{t}.\end{aligned}$$ Thus, it is 