i have that $a_{n}$ and $b_{n}$ are positives and reals and $0<b=\lim b_{n} < \infty$ and $a=\lim sup$ $ a_{n}$.
Lets consider the case when $\infty=\lim sup$ $ a_{n}$ then we can extract a subsequence such that $\lim a_{n_{k}}=a>0$. And since $\lim b_{n} >0$ we have that $\lim a_{n_{k}}b_{n_{k}}=\infty$.
my professor conclude like this but i dont understand why is that true.
As $a=\limsup a_n=\infty$, we can extract a divergent subsequence $a_{n_{k}}$ (one that keeps increasing for example, note that $a_n$ itself doesn't have to diverge, for example $a_n=0$ for even $n$ and $a_n=n$ for uneven $n$).
As $0<b=\lim b_{n} < \infty$, we also have that $\lim b_{n_k}=b$.
Now we get that for all $M>0$, we can find $N$ (such that $b_{n_k}$ is $\varepsilon=\frac{b}{2}$ close enough to $b$ and $a_{n_{k}}$ is larger than $2\frac{M}{b}$ for example) such that for all $k>N$ we have that $a_{n_{k}}b_{n_{k}}>2\frac{M}{b}(b-\varepsilon)=M$.
So we see that $\lim a_{n_{k}}b_{n_{k}}=\infty$.