Divergence by definition for the sequence $a_{n}=(-1)^n\frac{n}{n+5}$

Question

Consider the sequence $a_{n}=(-1)^n\frac{n}{n+5}$

How can I show that it diverges by definition, that is:

$\exists\space\epsilon>0;\forall\space n_{0}\in\mathbb{N},\exists\space n\ge n_{0};|a_{n}-l|\ge\epsilon$

Answer 1

Examine the difference between successive terms, $$ \left | (-1)^m\frac{m}{m+1}+(-1)^{m+2}\frac{m+1}{m+7}\right|= \left | \frac{m}{m+1}+\frac{m+1}{m+7}\right| =\frac{m}{m+1}+\frac{m+1}{m+7}\\ \geq \frac{m}{m+1}\geq\frac{1}{2} $$ whenever $m\geq 1$. Now, given $\epsilon=1/2$, can you see that for any $N\in \mathbb{N}$, we have for any $\ell\in \mathbb{R}$, either $$ |a_{N+1}-\ell|\geq1/2 $$ or $$ |a_{N}-\ell|\geq1/2 $$