Give an example of a sequence $(a_n)_n$ such that $|a_{n+1}/a_n|<1$ for all $n\in\mathbb{N}$, but where $(a_n)_n$ does not converge to $0$.
This question came up in a text book and I've no idea what the answer it, any help, sorry for not knowing how to type math equations!
Take $(a_n)_{n\in\mathbb{N}}$ s.t. $a_n=\frac{1}{n}+\frac{1}{2}$ f.a. $n\in\mathbb{N}$. You have that $a_{n+1}<a_n$ f.a. $n\in\mathbb{N}$, i.e. that $(a_n)_{n\in\mathbb{N}}$ is strictly monotone decreasing(check this). Thus
$$\frac{a_{n+1}}{a_n}<1$$
as all $a_n$ are positive but clearly $\lim_{n\to\infty}a_n=1/2$.
EDIT: To give some suggestion of how to come up with such examples: analyze you conditions. Note that in your case, if $a_n>0$ f.a. $n\in\mathbb{N}$, then $|\frac{a_{n+1}}{a_n}|=\frac{a_{n+1}}{a_n}$ and thus
$$|\frac{a_{n+1}}{a_n}|<1\Leftrightarrow\frac{a_{n+1}}{a_n}<1\Leftrightarrow a_{n+1}<a_n$$
Thus, under some assumptions, your condition is the same as requiring that the sequence decreases strictly monotone. From, this on, it might be simpler to come up with examples.