I am confused why the function $ f_n = n\chi_{[0, 1/n]}$ converges pointwise.
As I remember from my first analysis course, when we prove the pointwise convergence, we should fix $x$ first. Then, for $x \in [0,1/n]$, this function diverges. Doesn't it?
For the function to converge pointwise, $f_n(x) \to f(x)$ as $n \to \infty$ for every $x$. Then, shouldn't we say that $f_n$ does not converge pointwise? But, I found here that $f_n$ converge pointwise.
I think I am missing something. I appreciate if you give some help.
Best Answer
It doesn't converge at the point $0$, but for any other point $x\in[0,1]$ there is an $N\in\mathbb{N}$ such that $x>1/N$, so for all $n\geq N$ we have $x>1/n$ and hence $n\chi_{[0,1/n]}(x)=0$. Of course, if $x\notin [0,1]$ we trivially have $n\chi_{[0,1/n]}(x)=0$ for all $n$.
So the example should have omitted the point $0$.