I'm stuck with showing the existence of a limit, all I know essentially is that for $g_t \in \mathbb R$, $0 \leq g_{t+C} \leq g_t \leq A \leq \infty$ for all $t \in \mathbb R$ and some $C \in (0,\infty)$, does the limit necessarily exist under this condition? Clearly the sequence has a convergent subsequence, I was hoping that this could be strengthened to the whole sequence.
Also, if this is not true, then what about if $g_t \in \mathbb R$, $0 \leq g_{t+C} \leq g_t \leq A \leq \infty$ for all $t \in \mathbb R$ and for all $C \geq B > 0$.
Thanks.