I am solving CMI PHD entrance questions and one of the questions is as follows:
State whether true or false: If $a_n$ and $b_n$ are two positive sequences first one converging to $0$ and second one diverging to $\infty$,then the sequence $c_n=a_ne^{ib_n}$ converges to $0$.
I think the answer is yes because $e^{ib_n}$ has modulus $1$ i.e. it varies over the unit circle so $|c_n|=|a_n|\to 0$ as $n\to \infty$.So,whatever $b_n$ is,it does not matter,$c_n$ goes to $0$.But it is so easily done that I doubt whether my solution is correct or I am missing something.Can someone check the solution?