Let's define a family of functions indexed by positive integer $n$: $\{ f_n(\cdot)\}$, where $f_n: [0,1]\to\mathbb{R}$. Is there an example of $f_n$ so that the following condition holds for some $x_0$?
(Condition)
For small $n$, it could be that $f_n(x) \geq f_n(x_0)$ for some $x\ne x_0$, but there is $N \in \mathbb{N}$ such that $n \geq N \implies f_n(x) < f_n(x_0)$ for all $x \ne x_0$.
In other words, $x_0$ is the asymptotic maximizer of $f_n(x)$.