# Uniform Convergence of $e^{-nx}$ on $[0,1]$, $[a,1]$, $(0,1]$

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?

• I think the restriction on a should be $0<a\leq1$ – 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 $$[0, 1]$$.