Question 10
For every couple of real numbers $(a, b)$, consider the sequence of numbers $\{x_n\}$, $n = 0, 1, 2, \dots$, defined by:
$$
x_0 = a \quad \text{and} \quad x_{n+1} = x_n + b \sin x_n \quad \text{for every } n = 0, 1, 2, \dots
$$
Prove that:
1) for every real number $a$, the sequence $\{x_n\}$ corresponding to $(a, b)$ has a finite limit when $n$ tends to infinity. Find this limit.
2) for every number $b \gt 2$, there exists a real number $a$ such that the sequence $\{x_n\}$ corresponding to $(a, b)$ does not have a finite limit when $n$ tends to infinity.
$$
x_0 = a \quad \text{and} \quad x_{n+1} = x_n + b \sin x_n \quad \text{for every } n = 0, 1, 2, \dots
$$
Prove that:
1) for every real number $a$, the sequence $\{x_n\}$ corresponding to $(a, b)$ has a finite limit when $n$ tends to infinity. Find this limit.
2) for every number $b \gt 2$, there exists a real number $a$ such that the sequence $\{x_n\}$ corresponding to $(a, b)$ does not have a finite limit when $n$ tends to infinity.