The convergence part of the ratio test for real sequences says, roughly, that if the absolute values of the ratios of a given (non-zero) sequence $a_n$ are, for sufficiently large $n$, bounded below 1, then the given sequence $a_n$ converges to 0, since $a_n$ can be compared to a geometric sequence whose ratio is strictly less than 1.
However, the above scenario does not include the case in which the ratios $|\frac{a_{n+1}}{a_n}|$ are all strictly less than 1 but approach as close to 1 as one wants - eg., when their limit superior (or even their limit) is exactly 1.
I would like to have a counterexample in this situation.
More specifically, I would like to see an example of a sequence $a_n$ such that, for every $n\in\mathbb{N}$, we have $|\frac{a_{n+1}}{a_n}|<1$, the limit (or limsup) of $|\frac{a_{n+1}}{a_n}|$ is exactly 1, while $a_n$ does not converge.