Product of lim sups

Question

Suppose lim sup $a_n$ is finite, and $c_n \to c$

Prove that if $c \geq 0$ lim sup $a_n c_n$ = c lim sum $a_n$ and find a counterexample to this if $c <0$.

Is there a rule that the product of lim sups is equal to the lim sup of the product? Also, what counterexample will work here?

See also http://math.stackexchange.com/questions/947822/if-limsup-x-n-x-lim-y-n-y-x-n-y-n-0-then-does-limsup-x-n-y and http://math.stackexchange.com/questions/776517/product-of-limitsuperior-of-bounded-sequences — Martin Sleziak, Sep 27 '14 at 06:03

WWK · Accepted Answer · 2014-04-24T23:47:29.303

1

Let $c_n=-1$ for all $n$. Try to prove this:

$$\limsup_{n\to\infty}(-a_n)= = -\liminf_{n\to\infty}a_n $$

For general proof of positive $c$:

Since $c_n \rightarrow c$, then $\forall \alpha>0$, $\exists N>0$ such that $0<c-\alpha<c_n<c+\alpha$. Then $$(c-\alpha)\sup_{n>N} a_n <\sup_{n>N} c_n a_n< (c+\alpha)\sup_{n>N} a_n$$

edited Apr 24 '14 at 23:47

answered Apr 24 '14 at 22:59

WWK

1,370

Hmm, and how does this help with the general proof case for any $c_n$? – kiwifruit Apr 24 '14 at 23:30
I update the general case. – WWK Apr 24 '14 at 23:48
I already gave you the hint for counterexample. You only need to find a sequence such that $\liminf_{n\to\infty}a_n \neq \limsup_{n\to\infty}a_n$ – WWK Apr 25 '14 at 00:01
But by the general answer, lim sup $c_n a_n$ < c lim sup $a_n$ but we are asked to prove they are equal – kiwifruit Apr 25 '14 at 00:01
But I already gave you another direction. – WWK Apr 25 '14 at 00:04

Product of lim sups

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