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I'm reading the proof for $(1)$ of this paper, and I can't get the hang of how the author concludes the "hence we have that $L-\epsilon<\limsup b_n<L+\epsilon$", could anybody explain this?

I think he takes the $\limsup$ of each part, but even if that's the case, I don't get how to get that answer.

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I assume you've understood the proof up to

$${C\over n}+{(n-N)(L-\epsilon)\over n}<b_n<{C\over n}+{(n-N)(L+\epsilon)\over n}$$

We have $\limsup x_n\le \limsup b_n$ whenever $x_n\le b_n$. This is in Baby Rudin 3.19 so I won't write a proof out here. We also know that if $x_n$ converges to a limit, then that limit equals lim sup and lim inf.

Now, take the limits on the left and right above. The limit of $C/n$ is of course zero, $C$ being independent of $n$. The limit of $(n-N)/n$ is of course $1$, $N$ being again independent of $n$. $L-\epsilon$ and $L+\epsilon$ are independent of $n$. Thus, we get that the limit on the left is $L-\epsilon$ and the one on the right $L+\epsilon$.

Edited to add proof of Baby Rudin 3.19. We are working in the extended reals, with $\pm\infty$ included, per 3.16. We want show that if $s_n\le t_n$ for sufficiently large $n$, then $\limsup s_n\le \limsup t_n$ and the same for lim inf. Let's do lim sup.

Lim sup is by definition the sup of the set of subsequential limits. If $\limsup s_n=-\infty$ then the inequality we want is not constraining. Otherwise, let $s_{n_i}$ converge to $\limsup s_n$ (3.17(a)).

For large $i$, the sequence $t_{n_i}$ is bounded below by $\limsup s_n-\epsilon$ for any positive $\epsilon$. Because of the lower bound, either a subsequence of $t_{n_i}$ converges to a limit $L$ or one goes to $+\infty$. In the first case, the limit $L$ must be $\ge$ the limit of $s_{n_i}$ because of $s_n\le t_n$, and so $\limsup s_n\le L\le \limsup t_n$ as desired. In the second case, $\limsup t_n=+\infty$ and so the desired inequality is not constraining.

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  • $\begingroup$ Thanks for the reference! I'll look it up. If we had the sandwich theorem already, we wouldn't have to deal with $\lim$inf/sup right? $\endgroup$ Feb 1, 2016 at 4:05
  • $\begingroup$ Beat me to it. Note that Rudin's 3.19 proves that the same is true for $\liminf$ too. $\endgroup$
    – B. Freitas
    Feb 1, 2016 at 4:05
  • $\begingroup$ I am not sure if the sandwich theorem simplifies the proof. They have to fix $N$ in order to fix $C$ and get the other stuff to work. Maybe there's a shorter proof than theirs -- I'd have to think more about it. $\endgroup$
    – ForgotALot
    Feb 1, 2016 at 4:09
  • $\begingroup$ Yes, you're right, what they did is still needed. One last thing, I looked up $3.19$ on Baby Rudin but he just writes "We close this section with a theorem which is useful and whose proof is quite trivial." and leaves no proof, could you give me some tips on how to prove that? $\endgroup$ Feb 1, 2016 at 4:25

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