Proof of $\lim\sup(a_nb_n)\leq \lim\sup(a_n)\limsup(b_n)$

Question

Let $a_n>0$ and $b_n\geq 0$, then $\lim\sup(a_nb_n)\leq \lim\sup(a_n)\limsup(b_n)$

My attempt at a proof is as follows. Let $A_n=\sup\{a_n, a_{n+1},...\}$, $B_n=\sup\{b_n, b_{n+1},...\}$, and $C_n=\sup\{a_nb_n, a_{n+1}b_{n+1},...\}$.

Note: $a_mb_m \leq A_nB_n$ for all $m \geq n$.

Thus $\limsup(a_nb_n)=\lim C_n \leq \lim (A_nB_n) = (\lim A_n)(\lim B_n) = (\limsup a_n)(\limsup b_n).$

Is this proof even reasonable? That is, I feel like I'm missing something. — emka, Sep 17 '12 at 03:52
If you want to be super picky, to say $\lim(A_nB_n)=(\lim A_n)(\lim B_n)$, you need to first show the two individual limits exist. Of course this was probably done in defining what a limsup is in the first place. — Alex R., Sep 17 '12 at 03:57
the limit of products factors into the product of limits iff both individual limits $\lim A_n$ and $\lim B_n$ exist by themselves. As a contradiction, consider:
$1=\lim_{n\rightarrow\infty} \frac{n}{n}\neq (\lim n)(\lim \frac{1}{n})=undetermined$ — Alex R., Sep 17 '12 at 04:16
The missing argument is that $(A_n)$ and $(B_n)$ are nonnegative (here) nonincreasing (always) sequences. — Did, May 01 '14 at 08:32
See also lim sup inequality $\limsup ( a_n b_n ) \leq \limsup a_n \limsup b_n $ — Martin Sleziak, Mar 03 '17 at 11:01

score 2 · Answer 1 · answered Jun 25 '15 at 04:52

Missing pieces:

You need to assume the sequences are bounded. Otherwise the statement may be even nonsensical: how to interpret $1\le 0\cdot \infty$, for example?
$(A_n)$ and $(B_n)$ are nonnegative nonincreasing sequences; these properties imply the existence of the limits $\lim A_n$ and $\lim B_n$, which allows to conclude about the existence and the value of $\lim (A_nB_n)$.

You could have assumed $a_n\ge 0$ instead of $a_n>0$.

Proof of $\lim\sup(a_nb_n)\leq \lim\sup(a_n)\limsup(b_n)$

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