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?



Since $\displaystyle(\forall n\in\Bbb N):\left|f_n\left(\frac1{2n}\right)\right|=\frac n4$, you have $\displaystyle\sup_{x\in[0,1]}\left|f_n(x)\right|\geqslant\frac n4$. In other words, $\displaystyle\|f-f_n\|_\infty\geqslant\frac n4$ and, in particular, it is not true that $\displaystyle\lim_{n\to\infty}\|f-f_n\|_\infty=0$. So, the convergence is not uniform.
