Decomposition of Approximately Monotone and Convex Sequences

In this paper, we primarily deal with approximately monotone and convex sequences. We start by showing that any sequence can be expressed as the difference between two nondecreasing sequences. One of these two monotone sequences act as the majorant of the original sequence, while the other possesses non-negativity. Another result establishes that an approximately monotone(increasing) sequence can be closely approximated by a non-decreasing sequence. A similar assertion can be made for approximately convex sequence. A sequence \(\big_{n=0}^{\infty}\) is said to be approximately convex (or \(\varepsilon\)-convex) if the following inequality holds under the mentioned assumptions \begin{equation*} u_{i}- u_{i-1}\leq u_{j}-u_{j-1}+\varepsilon\quad\quad\mbox{ where}\quad\quad i,j\in\mathbb{N} \quad \mbox{with} \quad i