Let $\mathfrak{B}$ be the Borel $\sigma$-algebra on $\mathbb{R}$ and $m$ be the Lebesgue measure; let $1 \leq p < \infty$.
Show that $n^{-1/p} \chi_{[0,1]}$ does not converge to $0$ in $L^p$.
As far as I understand this, we need to show that:
$\left(\int_{\mathbb{R}} |n^{-1/p}\chi_{[0,1]}|^p \,dm\right)^{1/p} = \left(\int_{[0,1]}n^{-1}\,dm\right)^{1/p} $
doesn't converge to $0$ .
But in the limit as $n \rightarrow \infty$, $1/n \rightarrow 0$ and so wouldn't the right hand side tend to $0$? But this is precisely what I am trying not to show.
What am I doing wrong here? All help is much appreciated.