$newcommand{N}{mathbb{N}}

newcommand{R}{mathbb{R}}$

Show or find a counter example. Let $(a_{n})_{n in N} subseteq R$ be a sequence such that for all

(i) If there exits an $N in N$ and $q in R$, $q < 1$, such that:

$$left|frac{a_{n+1}}{a_{n}}right| leq q hspace{1cm} n in N, n geq N$$

then $displaystyle{lim_{n to infty}{a_{n}} = 0}$

(ii) If there exits an $N in N$ and $q in R$, $q leq 1$, such that:

$$left|frac{a_{n+1}}{a_{n}}right| < q hspace{1cm} n in N, n geq N$$

then $displaystyle{lim_{n to infty}{a_{n}} = 0}$