I have the following sequences of functions $f_n = n^{-1} \chi_{[0,n)}$. I want to show that this function converges uniformly and pointwise to $0$. For the uniform convergence, my proof is the following:

Fix some $\epsilon > 0$ and let $N =\frac{1}{\epsilon}$ such that for all $n \geq N$ we have that $|n^{-1}\chi_{[0,n)} -0 | = |n^{-1}\chi_{[0,n)}| \leq |\frac{1}{n}| \leq |\frac{1}{N}| < \epsilon$ and this is true for all $x \in \mathbb{R}$.

I understand that for showing pointwise convergence, we have to come up with the value of $N$ that depends on $\epsilon$ and $x$. My confusion comes up when thinking about this $N$. I have thinking about this for a long time but I haven't been able to come up with one. Any suggestions or hints to show this will be highly appreciated! Thanks!

  • 4
    $\begingroup$ Fun fact: somewhere in your lecture notes or textbook must be the lemma, observation, or theorem that uniform convergence implies pointwise convergence. $\endgroup$ – Clement C. Feb 28 '18 at 20:16
  • $\begingroup$ @ClementC you really made me laugh! :D $\endgroup$ – Netchaiev Feb 28 '18 at 20:21
  • $\begingroup$ @ClementC. I rather hope it is an observation and not a theorem $\endgroup$ – Tashi Walde Feb 28 '18 at 20:25
  • $\begingroup$ We don't "have to" : in general the N depends on $x$ for it is sufficient to prove pointwise convergence, but it does not have to (or, in other way to consider it, you can make it depends on $x$, since you can define $N(x):=N$ for every $x$ ; but that would be point...less) $\endgroup$ – Netchaiev Feb 28 '18 at 20:26

For $x <0,$ $f_n(x)=0/n=0$

For $x=0,$ $f_n (0)=1/n $

For $x>0$ and large enough $n,$

$f_n (x)=\frac {1}{n} $ since $x\in [0,n) $.

in all cases, $$\lim_{n\to+\infty}f_n (x)=0$$

There is pointwise convergence to zero function at $\Bbb R$

for the uniforme convergence, observe that $$|f_n (x)|\le \frac {1}{n}. $$


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.