I would like to construct a recurrence function $f$ over some sequence $v_0,v_1,v_2,\ldots$ such that $f(v_t) = c$ for some nonzero constant $c$ as $t\to\infty$. Specifically, I would like $f$ to model exponential decay of some state to some nonzero constant.
If $c=0$, this is easy, by letting $v_{t+1} = e^{-a}v_t$. Taking this, we get that $$ v_{t+1} = e^{-a}v_t = e^{-2a}v_{t-1} = \cdots = e^{-at}v_0$$
Therefore, we get that $v_t\to 0$. However, how would this work if $c\neq 0$? My initial idea was inspired by the fact that if $c=0$, then we essentially have that $v_t = e^{-a(t-1)}v_0$ which clearly goes to $0$. I wanted to get something of the form $v_t = e^{-a(t-1)}v_0 +c$ which goes to $0$, but I cannot figure out any recurrence function $f$ such that I can get that $f(v_t) = e^{-a(t-1)}v_0 +c$, where $f(v_t)$ has some dependence on $v_t$, other than just its time index.
How about $v_{t+1}=e^{-a}(v_t-c)+c$?