I want to ask 2 questions.
Question 1
Statement: If $a_n$ and $b_n$ are positive for all $n$, prove that $\limsup(a_nb_n)\leq(\limsup a_n)(\limsup b_n),$ provided the product on the right is not of the form $0\cdot\infty$
To prove this one, I set $\limsup a_n=\alpha, \limsup b_n=\beta.$
Then, $\exists n_0\in N s.t. a_n<\alpha+\epsilon$ for all $n\geq n_0$ and $b_n<\beta +\epsilon$ for all $n\geq n_0$.
So, $a_nb_n<(\alpha+\epsilon)(\beta+\epsilon)=\alpha\beta+(\alpha+\beta)\epsilon+\epsilon^2$
Since $\epsilon>0$ and is arbitrary one, I thought it is possible that $a_nb_n<\alpha\beta+\epsilon<\alpha\beta+(\alpha+\beta)\epsilon+\epsilon^2$. But I can't explain how it is possible logically.
Or I want to know if this is wrong approach.
Question 2
Statement: Let $a_n>0$ for all $n$. Prove that $\limsup \sqrt[n]{a_n}\leq \limsup {a_{n+1} \over a_n}$.
I can't handle this problem. Could you please give me some hints how to start the proof?
