Question 159
For a given real number $a$ and a positive integer $n$, prove that:
(1) there exists exactly one sequence of real numbers $x_0, x_1, \dots, x_n, x_{n+1}$, such that
$$
\begin{cases}
x_0 = x_{n+1} = 0, \\
\frac{1}{2}(x_{i+1} + x_{i-1}) = x_i + x_i^3 - a^3, \quad i = 1, 2, \dots, n;
\end{cases}
$$
(2) the sequence $x_0, x_1, \dots, x_n, x_{n+1}$ in (1) satisfies $|x_i| \le |a|, i = 0, 1, \dots, n+1$.
(1) there exists exactly one sequence of real numbers $x_0, x_1, \dots, x_n, x_{n+1}$, such that
$$
\begin{cases}
x_0 = x_{n+1} = 0, \\
\frac{1}{2}(x_{i+1} + x_{i-1}) = x_i + x_i^3 - a^3, \quad i = 1, 2, \dots, n;
\end{cases}
$$
(2) the sequence $x_0, x_1, \dots, x_n, x_{n+1}$ in (1) satisfies $|x_i| \le |a|, i = 0, 1, \dots, n+1$.