What does $n 1_{\{0 \le \omega \le \frac{1}{n}\}}$ converge to as $n \rightarrow \infty$

Question

Since $\frac{1}{n}$ has the same speed of convergence as $n$ what happens?

To be more specific the question in full: Consider a probability space $([0,1],\mathcal{B},\lambda)$ where $\lambda$ is the Lebesgue measure.

Let $X_n := n^{\frac{1}{p}}1_{\{0 \le \omega \le 1\}}$.

Show that $X_n \lim\nrightarrow^{L^P} 0$.

$L^p$ is L-p convergence. Didn't know how to do that in Latex.

Answer 1

By definition $X_n \to 0$ in $L_p$ $\Leftrightarrow$ $E|X_n-0|^p\to0$.

But here we have $E|X_n - 0^p| = E|n^{\frac{1}{p}}1_{\{0 \le \omega \le \frac1n\}}|^p = En1_{\{0 \le \omega \le \frac1n\}} = 1 \not\to0$.