Question 143
Let $f : [a, b] \to \mathbb{R}$ be an integrable function, and let $(a_n)_{n \ge 1}$ be a sequence of positive real numbers such that $\lim_{n \to \infty} a_n = 0$.
a) If $A = \{m \cdot a_n \mid m, n \in \mathbb{N}^*\}$, prove that any open interval of positive real numbers has elements from the set $A$.
b) If, for any $n \in \mathbb{N}^*$ and any $x, y \in [a, b]$, such that $|x - y| = a_n$, the inequality
$$
\left| \int_x^y f(t) \, dt \right| \le |x - y|
$$
holds, prove that:
$$
\left| \int_x^y f(t) \, dt \right| \le |x - y|, \quad \forall x, y \in [a, b].
$$
a) If $A = \{m \cdot a_n \mid m, n \in \mathbb{N}^*\}$, prove that any open interval of positive real numbers has elements from the set $A$.
b) If, for any $n \in \mathbb{N}^*$ and any $x, y \in [a, b]$, such that $|x - y| = a_n$, the inequality
$$
\left| \int_x^y f(t) \, dt \right| \le |x - y|
$$
holds, prove that:
$$
\left| \int_x^y f(t) \, dt \right| \le |x - y|, \quad \forall x, y \in [a, b].
$$