Find the point-wise limit of this sequence of function $\{f_n(x)\}$.

Question

Consider the sequence of function $\{f_n(x)\}$ in $[0,1]$ where , $$f_n(x)=\begin{cases}0 & \text{ if } x=0\\n^2x & \text{ if } x\in [0,\frac{1}{n}]\\-n^2x+n^2 & \text{ if } x\in (\frac{1}{n},\frac{2}{n}]\\0 & \text{ if } x\in [\frac{2}{n},1] \end{cases}$$

I have trouble to find the point-wise limit of this sequence of function $\{f_n(x)\}$.

I could not find the limit in the interval $(\frac{1}{n},\frac{2}{n}]$.

Can anyone help me to find the limit in this interval ?

Answer 1

There is no such thing as "the interval $[1/n, 2/n]$" when you are taking the limit of $n$ to infinity.

Hint:

For every $x\in (0,1)$, you can show that there exists some $N$ such that for each $n>N$, $x$ is not an element of $[1/n, 2/n]$.