Say we have two bounded sequences $a_{n}, b_{n}$, where $\lim_{n\to\infty}a_{n}=1$. I have to prove that $\limsup_{n\to\infty}a_{n}b_{n}=\limsup_{n\to\infty}b_{n}$. My idea was as following: First $b_{n}$ is bounded, so we can denote $\limsup b_{n}=L$. There exists a sub-sequence of $b_{n}$, $b_{n_{k}}$ that satisfies $\lim b_{n_{k}}=L$. Now we look at the sub-sequence of $a_{n}b_{n} - a_{n_{k}}b_{n_{k}}$. $\lim a_{n_{k}}=1$ as a sub-sequence of a converging sequence. So using limit arithmetic rules, $\lim a_{n_{k}}b_{n_{k}}=L=\limsup b_{n}$.
Now whats left is to prove that $\lim a_{n_{k}}b_{n_{k}}=\limsup a_{n}b_{n}$. Let's assume by contradiction that this is false. So there exists a sub-sequence $a_{n_{j}}b_{n_{j}}$ that satisfies $\lim a_{n_{j}}b_{n_{j}}=K, K>L$. $a_{n_{j}}$ converges to $1$ and $a_{n_{j}}b_{n_{j}}$ converges to $K$, so we can assume that $b_{n_{j}}$ converges to $K$, in contradiction to the assumption that $\limsup b_{n}=L$.
Is my argument correct or am I missing something?