Question 72
Let $a$ be a positive real number, $(a_n)_{n \ge 1}$ be a sequence of real numbers and $(x_n)_{n \ge 1}$ be the sequence defined by
$$
x_{n+1} = \left(1 - \frac{a}{n}\right) x_n + \frac{a_n}{n},
$$
where $x_1$ is an arbitrary real number. Prove that
$$
\lim_{n \to \infty} x_n = 0 \quad \text{if and only if} \quad \lim_{n \to \infty} \frac{a_1 + a_2 + \dots + a_n}{n} = 0.
$$
$$
x_{n+1} = \left(1 - \frac{a}{n}\right) x_n + \frac{a_n}{n},
$$
where $x_1$ is an arbitrary real number. Prove that
$$
\lim_{n \to \infty} x_n = 0 \quad \text{if and only if} \quad \lim_{n \to \infty} \frac{a_1 + a_2 + \dots + a_n}{n} = 0.
$$