# How to show a sequence is monotonically decreasing and a null sequence?

For example Let the sequence be $a_n=\frac{n+1}{n^2}$. I proved that $a_n$ is a null sequence by factoring out the $n^2$ .My question is how do i prove that it is monotonically decreasing? . Do i find the limit of the ratio of $\frac{a_{n+1}}{a_n}$ to infinity. Or do i show that $a_n$ is Cauchy ?

• What is a null sequence? – Umberto P. Feb 14 '17 at 19:58
• a sequence whose limit to infinty is 0 – asddf Feb 14 '17 at 19:58
• how do i prove that it is monotonically decreasing By definition, just prove that $a_{n+1} \le a_n\,$. – dxiv Feb 14 '17 at 19:58

By writing $$a_n=\frac1n+\frac1{n^2}, \qquad n\ge1,$$ one sees that $\{a_n\}$ is monotonically decreasing to $0$ being the sum of two monotonically decreasing sequences to $0$.
we compute $$a_{n+1}-a_n=\frac{n+2}{(n+1)^2}-\frac{n+1}{n^2}=\frac{n^2(n+2)-(n+1)^3}{n^2(n+1)^2}=-\frac{n^2+3n+1}{n^2(n+1)^2}<0$$