7

Proposition 19(v) of Section 1.5 page 23 in the book Real Analysis by Royden and Fitzpatrick, 4th edition (see link), says:

If $a_n \le b_n$ for all $n$, then $\lim\sup a_n \le \lim \inf b_n$.

However one can find a counter-example for a sequence $a_n = b_n$ which has different limsup and liminf. Is it a mistake, or am I missing something obvious?

Did
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user79963
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  • Yes, this is an error. And your explanation of why it's wrong is exactly correct; I can't imagine why someone downvoted this. – David C. Ullrich Sep 15 '15 at 15:28
  • I guess you have to swap $\liminf$ with $\limsup$. – egreg Sep 15 '15 at 15:29
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    I didn't vote up or down, but perhaps the downvote is because the poster links to an unabridged pdf of an unauthorized reprint of a copyrighted book. MSE tends to frown on such shenanigans. – Umberto P. Sep 15 '15 at 15:37
  • @UmbertoP. That would be solved by an edit, not certainly a downvote. – egreg Sep 15 '15 at 15:39
  • Of course it would. – Umberto P. Sep 15 '15 at 15:39
  • Are you sure that the link points to a legal copy of the book? If not, the link should be removed. – egreg Sep 15 '15 at 15:40
  • FYI: http://www.math.umd.edu/~pmf/err130405.pdf (This page is maintained by Fitzpatrick himself, so there is no legality concern here). This is in this list. – Ian Sep 16 '15 at 12:19
  • @UmbertoP. So Harvard/Curtis T McMullen is committing copyright infringement? It's not a link to some torrent site. It's a link to a Harvard webpage – BCLC Dec 03 '16 at 12:17
  • @Ian Is that supposed to be the whole book? Or just errata? The webpage now says 'Not Found

    The requested URL /~pmf/err130405.pdf was not found on this server.'

     – BCLC Dec 03 '16 at 12:18

3 Answers3

6

It seems you are right.

Take $a_n=0$ if $n$ is odd and $a_n=1$ if $n$ is even.

Take $b_n=a_n+0.5$

Then $\forall n$ $ a_n<b_n$ and $\limsup a_n=1$, $\liminf b_n=0.5$...

egreg
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Martigan
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5

You pinpointed the problem. Probably it's a typo and $$ \liminf\{a_n\}\le\limsup\{b_n\} $$ was meant.

egreg
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1

As others pointed out, it’s a typo, indeed.

What is true is that if $a_n\leq b_n$ termwise, then both $$\liminf_{n\to\infty}a_n\leq\liminf_{n\to\infty}b_n$$ and $$\limsup_{n\to\infty}a_n\leq\limsup_{n\to\infty}b_n$$ hold.

triple_sec
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