How we obtain this limit on an integral?

Question

For a squence $(u_n)\subset W^{1,p}(\mathbb{R}^N)$ and $(y_n)\subset \mathbb{R}^N$ we have that for a subseuqnce $|y_n|<M$ then

What we use to prove that $$ \limsup_{n\to+\infty}\int_{B_{r+M}(0)} |u_n(x)|^p dx\geq \limsup_{n\to+\infty}\int_{B_r(y_n)}|u_n(x)|^p dx $$

Thank you

Answer 1

We have $B_{r}(y_{n})\subseteq B_{r+M}(0)$: For $z\in B_{r}(y_{n})$, then $|z-y_{n}|<r$, so $|z|\leq|z-y_{n}|+|y_{n}|<r+M$, so $z\in B_{r+M}(0)$.

So \begin{align*} \int_{B_{r}(y_{n})}|u_{n}(x)|^{p}dx\leq\int_{B_{r+M}(0)}|u_{n}(x)|^{p}dx, \end{align*} now taking $\limsup_{n\rightarrow\infty}$ both sides.