Dear mathoverflowers, I have a question concerning the strong convergence in $L^p([0,T],X)$.

Let $X_1,X$ be two Banach spaces such that $X_1\subset X$ with compact embedding. Let $x_n(t)\in X_1$ be a bounded sequence in $X_1$ (this sequence converge strongly to $x(t)\in X$ for almost every $t\in [0,\infty)$). Moreover assume that $$ \|x_n(t)\|_X +\|x(t)\|_X \leq C $$ for almost every $t$.

Does this sequence converge strongly in $L^p([0,T],X)$? I think that, in fact, it does. I write my argument below. It seems quite easy, so I don't know if it is correct...

Let $T<\infty$ a fixed final time and define $\rho_n(t)=\|x_n(t)-x(t)\|^p_X$. Due to the compactness we have $\rho_n(t)\rightarrow 0$ for almost every $t$. We also have $\rho_n(t)\leq$C. I think that we can use now the dominated convergence theorem (taking $C$ as the dominating function and using the finiteness of the time interval) $$ \lim_n\int_0^T\rho_n(s)ds=0. $$

Therefore $$ \lim_n\|x_n-x\|_{L^p([0,T],X)}^p=\lim_n\int_0^T \|x_n(s)-x(s)\|^p_Xds=\lim_n\int_0^T\rho_n(s)ds=0, $$ and we obtain that $x_n$ converges strongly to $x$ in the Banach space $L^p([0,T],X)$.

Thank you in advance for your answer!!