Constructing a sequence $\{ f_n \} \subset C_{c}[0,1]$(the space of continuous functions with compact support in $[0,1]$) such that ..
$(a)$ $||{f_n}||_{\infty} \to \infty$ and $||{f_n}||_{1} \to 0.$
Here is my sequence: $f_n(x) = \begin{cases} n + nx, & \text{ if } 0 \leq x \leq \frac{1}{n^2},\\ 0, & \text{ if } x > \frac{1}{n^2}. \end{cases}$
My calculations said that I am correct in this sequence. right?
$(b)$ $f_n(x) \to 0$ for all $x$ and $||{f_n}||_{1} \to \infty.$
Here is a solution to it:
But I feel like the given function is not correct when $x = \frac{1}{2n},$ am I correct? If so, what is the intuition behind creating this function correctly? Does anyone one know a function that works?
Can anyone help me answer these questions please?