Finding a sequence $(x_n)$ such that $\lim_{n \to \infty} f_n(x_n)\neq \lim_{n \to \infty} f(x_n)$

Question

I have a sequence of functions given by

1. $f_n(x) = x^n$, on $D=(0,1)$

2. $f_n(x) = \begin{cases} 4n^2x , & 0 \leq x \leq \frac{1}{2n} \\ -4n^2x+4n ,& \frac{1}{2n} \leq x \leq \frac{1}{n} \\ 0 & \text{otherwise} \end{cases}$, on $D=[0,1]$.

How can I find a sequence $(x_n)$ in $D$ such that $$\lim_{n\rightarrow\infty} f_n(x_n) \neq 0 = \lim_{n\rightarrow\infty} f(x_n) ?$$

Answer 1

Look at where uniform convergence "breaks"; for $f_n$, this happens at $1$, and if you let $x_n=1-\frac{1}{n}$, you find that $$f_n(x_n)=\left(1-\frac{1}{n}\right)^n\to\frac{1}{e}\neq 0.$$ For the second function it is helpful to draw the graph; for fixed $n$ we get a piecewise linear function, and it has its maximum at $\frac{1}{2n}$. Setting $x_n=\frac{1}{2n}$, we have that $$f_n(x_n)=1\to 1\neq 0.$$