Here is an exercise: Let $\{a_{n}\}$ be a positive increasing sequence, can we prove that: $\liminf_{n\rightarrow\infty}\frac{a_{n}}{a_{n-k}}\leq \liminf_{n\rightarrow\infty}a_{n}^{k/n}$? Could someone give me some hints?
Asked
Active
Viewed 67 times
1
-
3Let $L = \liminf\limits_{n\to\infty} \frac{a_n}{a_{n-k}}$. If $L = 0$, you're done. If $L > 0$, for every $\varepsilon > 0$ there is an $n_\varepsilon$ such that $$\frac{a_n}{a_{n-k}} > L-\varepsilon$$ for all $n \geqslant n_\varepsilon$. – Daniel Fischer Apr 15 '14 at 14:51
-
1The inequality $\liminf(a_{n+1}/a_n) \le \liminf((a_n)^{(1/n)})$ seems related, so looking at a proof of that one might help: http://math.stackexchange.com/questions/69386/inequality-involving-limsup-and-liminf – Martin Sleziak Apr 17 '14 at 08:13
-
Please, try to make the title of your question more informative. E.g., Why does $a<b$ imply $a+c<b+c$? is much more useful for other users than A question about inequality. From How can I ask a good question?: Make your title as descriptive as possible. In many cases one can actually phrase the title as the question, at least in such a way so as to be comprehensible to an expert reader. You can find more tips for choosing a good title here. – Martin Sleziak Apr 17 '14 at 08:16