Use interpolation inequality to improve strong convergence

Question

I'm considering a sequence $f_{\epsilon}\in L^{1}(\Omega)\cap L^{\infty}(\Omega)$. Then by interpolation inequality, I have for all $1<p<\infty$ with $0<\theta<1$ such that $$\lVert f_{\epsilon}\lVert_{p}\leq C\lVert f_{\epsilon}\rVert_{1}^{1-\theta}\lVert f_{\epsilon}\rVert_{\infty}^{\theta}.$$ Now assume we have $f_{\epsilon}\to f$ in $L^{1}$ strongly and $f_{\epsilon}$ is $\epsilon-$uniformly bounded in $L^{\infty}$, then I think \begin{align*} \lVert f_{\epsilon}-f\lVert_{p}&\leq C\lVert f_{\epsilon}-f\rVert_{1}^{1-\theta}\lVert f_{\epsilon}-f\rVert_{\infty}^{\theta}\\ &\to0 \end{align*} because the first term goes to zero and the second term is bounded. Hence, $f_{\epsilon}\to f$ in $L^{p}$ strongly. Does this make sense?
If this is the case, then I think for gerneral interpolation space for example $(X_{0},X_{1})_{\theta}=X$ with $\theta\in(0,1)$, if I have my sequence uniformly bounded in e.g. $X_{1}$ and strong convergence in $X_{0}$, then it is also strong convergence in $X$. Does this also make sense?