Suppose that ${a_n}$ and $b_n$ is decreasing sequence such that $a_0=A$, $lim_{n->\infty}a_n=0$ and $b_0=B$, $lim_{n->\infty}b_n=0$.
For fix n, we can construct n dimension linear equation system such that
\begin{eqnarray} a_n x_n+a_{n-1}x_{n-1}+\cdots+a_0x_0&=&b_0\\ a_{n}x_{n-1}+\cdots+a_1x_0&=&b_1\\ \vdots\\ a_{n}x_0&=&b_n \end{eqnarray}
then solutions are uniquely determined. If n increases to $\infty$, i want know whether the solutions are bounded. If so, how can it be proved? If not, which condition is added to satisfy the bounded solution?
Thanks in advances.