SEEMOUS 2021 Problem 1

Let $f: [0, 1] \to \mathbb{R}$ be a continuous strictly increasing function such that $$\lim_{x \to 0^+} \frac{f(x)}{x}=1.$$ (a) Prove that the sequence $(x_n)_{n \ge 1}$ defined by $$x_n=f \left(\frac{1}{1} \right)+f \left(\frac{1}{2} \right)+\cdots+f \left(\frac{1}{n} \right)-\int_1^n f \left(\frac{1}{x} \right) \mathrm dx$$ is convergent. (b) Find the limit of the sequence $(y_n)_{n \ge 1}$ defined by $$y_n=f \left(\frac{1}{n+1} \right)+f \left(\frac{1}{n+2} \right)+\cdots+f \left(\frac{1}{2021n} \right).$$