Question 153
Let $f:[a, b] \longrightarrow \mathbb{R}$ be an integrable function, and let $(a_{n})_{n \geq 1}$ be a sequence of strictly positive real numbers such that $\lim_{n \rightarrow \infty} a_{n}=0$.
a) If $A=\{m \cdot a_{n} \mid m, n \in \mathbb{N}^{*}\}$, prove that every open interval of strictly positive numbers contains elements of $A$.
b) If for every $n \in \mathbb{N}^{*}$ and all $x, y \in [a, b]$ with $|x-y|=a_{n}$ the inequality $\left|\int_{x}^{y} f(t) d t\right| \leq |x-y|$ holds, prove that:
$$
\left|\int_{x}^{y} f(t) d t\right| \leq |x-y|, \quad \forall x, y \in [a, b]
$$
a) If $A=\{m \cdot a_{n} \mid m, n \in \mathbb{N}^{*}\}$, prove that every open interval of strictly positive numbers contains elements of $A$.
b) If for every $n \in \mathbb{N}^{*}$ and all $x, y \in [a, b]$ with $|x-y|=a_{n}$ the inequality $\left|\int_{x}^{y} f(t) d t\right| \leq |x-y|$ holds, prove that:
$$
\left|\int_{x}^{y} f(t) d t\right| \leq |x-y|, \quad \forall x, y \in [a, b]
$$