Question

Prove that the sequence {a_n /n } is convergent where { a_n } is a bounded sequence.

Answer 1

Suppose that the sequence $a_n$ is bounded. It means that there is $M>0$ such that $|a_n|<M$ $\forall n$. Therefore, $|\frac{a_n}{n}|<\frac{M}{n}$. Since $\frac{M}{n}\rightarrow0$, $n\rightarrow\infty$, $|\frac{a_n}{n}|\rightarrow0$, $n\rightarrow\infty.$