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?

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

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.

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