I am a bit embarassed about this question, but I’ll go ahead anyway.
Let $\Omega$ be an open bounded space and consider the Sobolev space $W_0^{1,p}(\Omega)$ endowed with its usual norm $\|\cdot\|$. Let $(u_n)_n\subset W_0^{1,p}(\Omega)$ be a sequence such that $$(\| u_n\|)_n \leq \beta,$$ for a positive constant $\beta$ which does not depend on $n$.
My question is: for $\alpha>\beta$, we can say that $n_0\in\mathbb{N}$ exists such that $$\| u_n\| \leq\alpha\quad\mbox{ for all } n\geq n_0?$$
I am sorry for my probably stupid question, but I hope someone could help.
Thank you in advance!
Of course: if $\|u_n\|\leq \beta$ for all $n\in\mathbb{N}$ and $\alpha>\beta,$ then $$\|u_n\|\leq \beta<\alpha$$ for all $n\in\mathbb{N}$. So, it is true for any $n_0\in\mathbb{N}$.
Saying $(\|u_n\|)_n\leq\beta$ is the same thing as saying $\|u_n\|\leq\beta$ for all $n\in\mathbb{N}.$