I'm currently working with two sequences of functions, $u$ and $u_n$, where $u, u_n \in L^{2}(0,T,H_{0}^{1}(\Omega)) \cap C([0,T],L^{2}(\Omega))$ and their derivatives $u', u_n' \in L^{2}(0,T,H^{-1}(\Omega))$. Given the following condition:
$$ \lim_{n\to\infty}\left[\|u{}_{n}(t_{s})-u(t_{s})\|_{L^{2}(\Omega)}^{2}+2\int_{0}^{t_{s}}\|u_{n}(t)-u(t)\|_{H_{0}^{1}(\Omega)}^{2}dt\right]=0\quad t_{s}\in[0,T] $$
it can be shown that $u_{n}(t)\to u(t)$ in $L^{2}(0,T,H_{0}^{1}(\Omega))$, and furthermore,
$$ \lim_{n\to\infty}\int_{0}^{T}\left[u_{n}(t)-u(t)\right]\phi'(t)dt=0 $$
My goal is to demonstrate that $u_{n}'\to u'$ in $L^{2}(0,T,H^{-1}(\Omega))$. I'm considering whether the following inequality needs to be proven:
$$ \sup_{v\in H_{0}^{1}(\Omega),\|v\|=1}\left|\left<u_{n}'(t)-u'(t),v\right>\right|\leq C\|u_{n}(t)-u(t)\|_{H_{0}^{1}(\Omega)} $$
By applying the Poincarré inequality, I was able to derive:
\begin{align*} \int_{0}^{T}\left(u_{n}'(t)-u'(t),v\phi(t)\right)_{L^{2}(\Omega)}dt & =\int_{0}^{T}\left(u_{n}(t)-u(t),v\phi'(t)\right)_{L^{2}(\Omega)}dt\\ & \leq\int_{0}^{T}\phi'(t)\|u_{n}(t)-u(t)\|_{H_{0}^{1}(\Omega)}\|v\|_{H_{0}^{1}(\Omega)}dt \end{align*}
However, at this point, I find myself stuck. Are there any additional conditions or techniques I could apply to help prove that $u_{n}'\to u'$ in $L^{2}(0,T,H^{-1}(\Omega))$? Any help or insight into this matter would be greatly appreciated.
Thank you in advance for your time and assistance.