While self-studying I stumbled upon a problem, Exercise 1.4.1 from Katznelson's An introduction to Harmonic Analysis asking that:
Given a sequence $\{\omega_n\}$ of positive numbers such that $\omega_n \to 0$ as $n \to \infty$, show that there exists a sequence $\{a_n\}$ which tends to $0$ at infinity satisfying $$a_{n-1}+a_{n+1}-2a_n \geq 0 \quad \text{and} \quad a_n \geq \omega_n\ \text{for all $n$.}$$
I've managed to find out that we can bound $\{\omega_n\}$ by a decreasing sequence converging to $0$, hence without loss of generality we may assume $\{\omega_n\}$ decreases to $0$.
So if we can somehow show that a sequence decreasing to $0$ can be bounded by a convex sequence which decreases to $0$ then we are done. However I am stuck right here, without any clue to proceed. Am I on the right track? Any help would be greatly appreciated.