 Bounded solution of a second-order constant-coefficient recurrence

For a linear recurrence relation with two distinct characteristic equation of a linear recurrence roots $r_1,r_2$, the solution is $u_n=A{r}_1^n+B{r}_2^n$. If $|r_1|<1<|r_2|$, boundedness for $n\to \infty$ forces $B=0$, and $u_0$ fixes $A$. This selects the decaying discrete mode in a finite difference method for a boundary problem on an unbounded interval. Roots on the unit circle require separate treatment; a repeated root there can contribute an unbounded factor $n$.