Let $\{a_n\}_{n=1 }^{\infty}$ and $\{b_n\}_{n=1}^{\infty} $ be two sequences in $\mathbb{R}$, with the first sequence convergent . Prove that $$ \limsup\limits_{n\to \infty} a_n b_n =\lim\limits_{n\to\infty} a_n \limsup\limits_{n\to\infty} b_n$$
I tried following: $ \limsup\limits_{n\to \infty} a_n b_n \leq \limsup\limits_{n\to\infty} a_n \limsup\limits_{n\to\infty} b_n$. Since $\{a_n\}$ is convergent, $\limsup\limits_{n\to\infty} a_n =\lim\limits_{n\to\infty} a_n $ gives one inequality along with one of the property of the limsup of the product of two sequences i.e $ \limsup\limits_{n\to \infty} a_n b_n \leq \lim\limits_{n\to\infty} a_n \limsup\limits_{n\to\infty} b_n$. I want to prove $ \limsup\limits_{n\to \infty} a_n b_n \geq \limsup\limits_{n\to\infty} a_n \limsup\limits_{n\to\infty} b_n$.
Can anyone help me on this?