0
$\begingroup$

Let {$f_n$} be a sequence of functions defined on $\mathbb R$ by $f_n(x) : = e^{-nx}$.

Does it converge uniformly on $[0,1]$? Does it converge uniformly on $[a,1]$ with $0 < a \leq 1$ ? Does it converge uniformly on $(0,1]$ ?

Clearly, {$f_n$} converges pointwise to $f(x)$ defined by

\begin{cases} 1 & x= 0 \\ 0 & x\in (0,1] \end{cases}

How do I prove that it doesn't converge uniformly on any of the intervals?

$\endgroup$
  • 1
    $\begingroup$ I think the restriction on a should be $0<a\leq1$ $\endgroup$ – guy3141 May 15 at 22:20
3
$\begingroup$

The pointwise limit is not continuous on $[0, 1]$, so $f_n$ couldn't converge to $f$ uniformly on $[0, 1]$.

This should provide an answer to the first and a hint to the other two.

| cite | improve this answer | |
$\endgroup$

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.