Question

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

Answer 1

As $a_n$ is bounded, we know that there exists a positive real number $M$ such that $|a_n|\leq M$ for all $n\in \mathbb{N}$. We will prove that the sequence $(\frac{a_n}{n})_{n\in\mathbb{N}}$ converges to zero, let us fix $\varepsilon>0$. We know that there exists $N\in\mathbb{N}$ such that for all $n\geq N$, $\frac{M}{n}< \varepsilon$ (it is enough to take $N=\text{smallest integer greater than } M/\varepsilon$). We have then an estimate for any $n\geq N$ :

$|\frac{a_n}{n}|\leq |\frac{M}{n}|<\varepsilon$

Therefore, $(a_n/n)_{n\in\mathbb{N}}\to 0$.