Question about $\lim_{n\to\infty} n|\sin n|$

Question

I have a question regarding this limit (of a sequence): $$\lim_{n\to \infty} n|\sin(n)|$$ Why isn't it infinite? The way I thought this problem is like this-$|\sin(n)|$ is always positive, and n tends to infinity, so shouldn't the whole limit go to infinity? What is the right way to solve this and why is my idea wrong?

Answer 1

The limit of the sequence $\{n\left|\sin n\right|\}_{n\geq 0}$ as $n\to +\infty$ does not exist. Obviously $\left|\sin n\right|$ is arbitrarily close to $1$ for infinite natural numbers, making the $\limsup=+\infty$. On the other hand, if $\frac{p_m}{q_m}$ is a convergent of the continued fraction of $\pi$ we have $$ \left|p_m -\pi q_m\right|\leq \frac{1}{q_m} $$ and since $\sin(x)$ is a Lipschitz continuous function, the $\liminf$ is finite, by considering $n=p_m$.