# real analysis – True and False Sequence convergence criterions

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