Show that $\lim_{n\rightarrow \infty} (1-\omega(\frac{1}{n}))^n=0$ and $\lim_{n\rightarrow \infty} (1-o(\frac{1}{n}))^n=1$

Asked Feb 04 '16 at 20:04

Active Feb 04 '16 at 20:04

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Could you help me to show that

(1) $\lim_{n\rightarrow \infty} (1-\omega(\frac{1}{n}))^n=0$

(2) $\lim_{n\rightarrow \infty} (1-o(\frac{1}{n}))^n=1$

where $o(\cdot)$ is little $o$ notation described here and $\omega(\cdot)$ is little $\omega$ notation described here

asked Feb 04 '16 at 20:04

2

I think you need to be careful: your $\omega(1/n)$ should also be bounded above in magnitude by $1$. For instance just the sequence $-2$ is $\omega(1/n)$ but that certainly violates your first equality. – Ian Feb 04 '16 at 20:05
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Little-$o$ is handled in this answer from me: http://math.stackexchange.com/a/1451245/7933 – Thomas Andrews Feb 04 '16 at 20:07
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Actually, I made a minor error: your $\omega(1/n)$ needs to be between $0$ and $2$, and it can't have $2$ as a limit point. Otherwise everything's OK. – Ian Feb 04 '16 at 20:23

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