- #1

Rectifier

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**The problem**

I am trying determine wether ##f_n## converges pointwise or/and uniformly when ## f(x)=xe^{-x} ## for ##x \geq 0 ##.

**Relevant equations**

##f_n## converges

__pointwise__if ## \lim_{n \rightarrow \infty} f_n(x) = f(x) \ \ \ \ \ ## (1)

##f_n## converges

__uniformly__if ## \lim_{n \rightarrow \infty} || f_n - f || = 0 \ \ \ \ \ ## (2)

**The attempt**

##f_n(x) = f(nx) =nxe^{-nx} ##

Pointwise? (1)

## f(x) = \ \lim_{n \rightarrow \infty} f_n(x) = \lim_{n \rightarrow \infty} nxe^{-nx} = \lim_{n \rightarrow \infty} \frac{nx}{e^{nx}} = 0##

Uniformly? (2)

## 0 = \ \lim_{n \rightarrow \infty} || f_n - f || = \lim_{n \rightarrow \infty} || nxe^{-nx} - xe^{-x} || ##:

I am not sure how to continue from here and wether the last step was correct:

## \lim_{n \rightarrow \infty} || nxe^{-nx} - xe^{-x} || = 0 ##

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