Find a sequence $f_n$ so that $\int_0 ^1 |f_n(x)| = 2$ and $\lim_{n \to \infty} f_n(x) = 1$. The entire problem reads:
(a) Find a sequence $f_n: [0,1] \rightarrow \mathbb{R}$ such that $\int_0 ^1 |f_n(x)| = 2$ for all $n \in \mathbb{N}$ and $\lim_{n \to \infty} f_n(x) = 1$ for all $x \in [0,1]$. (b) If the $f_n$ are as in part (a), then prove $\lim_{n \to \infty} \int_0 ^1 |f_n(x) -1| dx = 1$. 
I'm having trouble finding the proper $f_n$ to fit this situation. Any help is appreciated. Once I find $f_n$, I may need tips for part (b) as well, but I think I could figure that part out given (a). Thanks in advance. 

Choose $$f(x):=4n^2x+1 \text{ for }x\in \left[0,\frac{1}{2n}\right]\\
f(x):=-4n^2x+1+4n \text{ for }x\in \left[\frac{1}{2n},\frac{1}{n}\right]\text{and }f(x)=1\text{ otherwise}.$$
