Suppose $u:{\bf R}^n\to{\bf R}$ is some continuous function and $\{u_j\}$ is a sequence of continuous functions on ${\bf R}^n$ such that $$ u_j\to u\quad\text{locally uniformly in }{\bf R}^n\ \text{as }j\to\infty \tag{1} $$ Suppose $u$ has a strict local maximum at some point $x_0\in{\bf R}^n$, i.e., $$ u(x)<u(x_0)\tag{2} $$ for all points $x$ sufficiently close to $x_0$ with $x\neq x_0$. By (2), for each (fixed) sufficiently small $R>0$, we have $$ \max_{\partial B_R} u < u(x_0) \tag{3} $$ where $B_R$ is the closed ball of radius $R$ centered at $x_0$. In view of (1), there exists some postive integer $N(R)$ (may depend on $R$) such that $$ \max_{\partial B_R} u_j < u_j(x_0)\quad\text{for all } j>N(R)\tag{4} $$ Consequently, $u_j$ (with $j>N(R)$) attains a local maximum at some point in the interior of $B_R$.
If we replace $R$ by sequence of radii tending to zero, we can conclude that for each sufficiently large $j,$ there exists a point $x_j$ at which $u_j$ has a local maximum and $x_j\to x_0$ as $j\to\infty$.
The argument above is tailored and paraphrased from a long introduction of viscosity solution of Hamilton-Jacobi equations in Evans's Partial Differential Equations (Chapter 10.1).
Question: Would anyone elaborate the very last sentence in the argument above?
I can only see the following is true:
if we replace $R$ by sequence of radii tending to zero, we can conclude that there exists a subsequence $u_{j_m}$ with the property that $u_{j_m}$ has a local maximum at some $x_{j_m}$ and $x_{j_m}\to x_0$ as $m\to\infty$.
Here is the original argument in Evans's book:
Obtaining $x_j$ for all sufficiently large $j$'s and not only for a subsequence is not harder. Let me summarize: for every $R>0$ there is $N(R)$ such that for each $j \ge N(R)$, $u_j$ attains a local maximum at some point in the interior of $B_R$.
Now for a fixed sequence $R_k \to 0$, we can assume that the corresponding sequence $N(R_k)$ is increasing. For each $k$ and $N(R_k) \le j < N(R_{k+1})$, $u_j$ attains a local maximum at some point $x_j$ in the interior of $B_{R_k}$. It is evident that $x_j \to x_0$.