I am doing the exercise 8.2 from the brezis' book of Functional Analysis and PDEs, And I'm having trouble with the second question, which says:
Construct a bounded sequence $(u_n)$ in $W^{1,1}((0,1))$ that admits no sub-sequence con- verging in $L^\infty ((0,1))$.
In the book he suggest to work with the following sequence:
$$u_n(x)=\begin{cases} 0, & 0 \leq x \leq \frac{1}{2} \\ n(x-\frac{1}{2}), & \frac{1}{2} \leq x \leq \frac{1}{2}+\frac{1}{n} \\ 1, & \frac{1}{2}+\frac{1}{n} \leq x \leq 1 \end{cases}$$ This sequence is obviously in $W^{1,1}((0,1))$, and it has no limits, but how can I proof that it can't has any converging sub-sequence in $L^\infty ((0,1))$? Any hint please?