Why is this sequence not uniformly convergent?

Question

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?

Answer 1

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.

Answer 2

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.

Answer 3

By definition of convergence of a sequence in a normed(or in general metric space)space the sequence (fn) cannot converge to f because norm (here it is sup-norm) of (fn - f) >= 1/4 for all n.