Let $s\in (0, 1)$ and consider the fractionl Sobolev space $H^s(\mathbb R)$ (see e.g. https://www.sciencedirect.com/science/article/pii/S0007449711001254).
Let $(A, ||\cdot||)$ be a Banach space and suppose that $A\subset H^s(\mathbb R)$, with the embedding being continuous, namely there exists $c$ depending on $s$ such that $||u||_{H^s}\le c||u||$ for any $u\in A$.
Assume that $(u_n)\subset A$ is bounded in $A$. The question is: Is it possible to conclude that, up to a renamed subsequence, $$u_n\to u\quad\text{ in } L^p_{loc}(\mathbb R) \text{ for some $p$}?$$
There is any $p$ for which it holds independently on the choice of $s\in (0, 1)$? (I mean $s>1/2$ or $s<1/2$) Could you please provide a reference?
Thank you in advance.