Non-uniformly integrable function

Question

So I need to show that the sequence $f_{n} = nI_{(0,1/n)} - nI_{(1/n,2/n)}$ is integrable in the limit but not uniformly integrable. I was able to show the first part and know that $g_n = nI_{(0,1/n)}$ is not uniformly integrable by consider $$ \sup_{n}\int_{[|g_n| > \alpha]}|g_n|d\mu $$ which equals 1 when $\alpha < n$. However, I don't know how to show for $f_n$.

Answer 1

$|f_n|=nI_{(0,1/n)} + nI_{(1/n,2/n)} >\alpha$ for $0<\alpha<n$ on both intervals $(0,1/n)$ and $(1/n,2/n)$, so $$\int_{[|f_n| > \alpha]}|f_n|d\mu=2$$