TITLE: If $\lim_{n\to \infty}a_n = a\in \mathbb{R}$ . Prove that $\limsup_{n\to \infty}a_n x_n=a\limsup_{n\to \infty}x_n$ .
QUESTION [5 upvotes]: Note: $x_n$ is a sequence which is not necessarily convergent.  
The following was my attempt.
Since $\lim_{n\to \infty}a_n=a$ then $\limsup_{n\to \infty}a_n=a$ .
Also $\sup(a_nx_n)=\sup(a_n)\sup(x_n)$.
Therefore, $\limsup_{n\to \infty}a_nx_n=\limsup_{n\to \infty}a_n\limsup_{n\to \infty}x_n=a\limsup_{n\to \infty}x_n$.  
I am not sure if this is correct. So could someone please show me how it is done. Thanks

REPLY [2 votes]: Let us work with the definition:
$$\limsup_{n\to\infty} x_n = \lim_{n\to\infty} \sup_{k\ge n} x_k.$$
(There are several equivalent definitions of limit superior. This one seems to be very often introduced as the first of them.)
We are trying to show that if $x_n>0$, $a>0$ and $\lim\limits_{n\to\infty} a_n=a$ then
$$\limsup_{n\to\infty} a_nx_n = a\limsup_{n\to\infty} x_n.$$
(It has been already pointed out in other answers and comments that the claim is not true without some assumptions on the sequences $a_n$, $x_n$; i.e., in the way it is stated in the question.)
For any given $\varepsilon>0$ we have some $n_0$ such that $k>n_0$ implies $$a-\varepsilon \le a_k \le a+\varepsilon.$$
We will only work with $\varepsilon<a$, so that we have $a-\varepsilon >0$. 
Then we have
\begin{gather*}
a-\varepsilon \le a_k \le a+\varepsilon\\
(a-\varepsilon)x_k \le a_kx_k \le (a+\varepsilon)x_k
\end{gather*}
for $k>n_0$ and
\begin{gather*}
\sup_{k\ge n} (a-\varepsilon)x_k \le \sup_{k\ge n} a_kx_k \le
\sup_{k\ge n} (a+\varepsilon)x_k\\
(a-\varepsilon)\sup_{k\ge n} x_k \le \sup_{k\ge n} a_kx_k \le
(a+\varepsilon)\sup_{k\ge n} x_k
\end{gather*}
Applying limit $n\to\infty$ to the above equation gives
$$(a-\varepsilon)\limsup_{n\to\infty} x_n \le \limsup_{n\to\infty} a_nx_n \le (a+\varepsilon)\limsup_{n\to\infty} x_n.$$
Since the above inequality is true for every $\varepsilon$ such that $0<\varepsilon<a$, we get that
$$\limsup_{n\to\infty} a_nx_n = a\limsup_{n\to\infty} x_n.$$
Notice that the above argument also works for $\limsup x_n=0$.
The claim from the title also works for $a=0$, but in this case we would have to disallow $\limsup x_n=+\infty$. (And the proof would be different.)
Similarly, the claim is true also in the case $\lim a_n=a=+\infty$. But now we would have to disallow $\limsup x_n=0$. (In the other words, we have to avoid indeterminate forms $0\cdot\infty$ and $\infty\cdot0$ in the product $a\cdot\limsup x_n$.)