How the Sobolev embedding used in this case

Question

Assume $\{u_k\}\subset H^1(S)$ and $$u_k \rightharpoonup u_0 \quad in \quad H^1(S), $$ where “$\rightharpoonup$” means “converges weakly”, $S$ be the unit circle parameterized by angle $\theta$ and $H^1(S)$ means Hilbert space equipped with norm $$\|u\|=\left\{\int_S[(u^{\prime})^2+u^2]d\theta\right\}^{\frac{1}{2}}.$$ Then by Sobolev imbedding, we have $$u_k\rightarrow u_0 \quad in \quad C^{\beta}, \forall \beta<\frac{1}{2}.$$ My question is how the Sobolev Embedding Theorem used in this step?
The Sobolev Imbedding Theorem is equal to Sobolev Embedding Theorem ?(This answer is in the comment)
Why the $\beta <\frac{1}{2}$ ? Thanks in advance!

Answer 1

If $\{u_n\}$ converges weakly to $u$ in the $H^1-$norm, then it converges strongly in the $H^s-$norm, for every $s<1$, $s\in\mathbb R$.

So, if $\beta<1/2$, then there exists an $s<1$, such that $ H^s(S)\subset C^\beta(S)$, i.e. $$ s=\beta+½. $$