Example where $\|\,f_n\|_\infty\to \infty$, but $\|\,f_n\|_1\to 0$

Question

Please help me to solve the following problem that is in the Lebesgue integral discussion

Give an example of a sequence $\,\,f_n : [0, 1] \to \Bbb R$ of continuous functions such that $\,\,\|f_n\|_\infty \to \infty$ but $\int_0^1\lvert\, f_n\rvert\,d\lambda \to 0$ as $n\to\infty$.

Original screenshot

Answer 1

Let $$ f_n(x)=\max\left\{n,n^{-1}x^{-1/2}\right\}. $$

Then $\|\,f_n\|_\infty=n$ while $\|\,f\|_1<\dfrac{1}{2n}$.