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.

cannotbe invertible, as the two spaces are not isomorphic. $\endgroup$