Why is this sequence not uniformly convergent? In this problem is explained that $f_n(x)$ is pointwise convergent, however not uniformly convergent. The explanation why is not unifromly convergent is also given. However I cannot understand it, when I use the  theorem below I get that limit of $f_n - f = 0$
Could maybe someone give me more detail answer why the sequence is uniformly convergent?



First, you have to determine the pointwise limit. Let $x\in[0,1]$. For $n>1/x$, $f_n(x)=0$, so the pointwise limit is $0$.
As the explanation shows, we have $\|f_n\|_\infty=n/4$. Thus, $\lim_{n\rightarrow\infty}\|f_n\|_\infty=\lim_{n\rightarrow\infty}n/4=\infty$ and using the theorem you quoted, the limit of the suprema diverging is equivalent to $f_n$ not converging uniformly.
