In this paper, our primary objective is to study a possible decomposition of an approximately convex sequence. For a given \(\varepsilon >0\) , a sequence \(\big <u_n\big >_{n=0}^{\infty }\) is said to be \(\varepsilon \) -convex if, for any \(i,j\in \mathbb {N}\) with \(i<j\) , there exists an \(n\in ]i,j]\cap \mathbb {N}\) such that the following discrete functional inequality holds: \(\begin{aligned} u_i-u_{i-1}-\dfrac{\varepsilon }{n-i}\le u_j-u_{j-1}. \end{aligned}\) We show that such a sequence can be represented as the algebraic sum of a convex and a controlled sequence which is bounded in between \(\left[ -\dfrac{\varepsilon }{2}, \dfrac{\varepsilon }{2}\right] .\) On the other hand, for any \(i,j\in \mathbb {N}\) with \(i<j\) , if a sequence \(\big <u_n\big >_{n=0}^{\infty }\) satisfies the inequality \(\begin{aligned} \left| \big (u_i-u_{i-1}\big )-\big (u_j-u_{j-1}\big )\right| \le \dfrac{\varepsilon }{n-i}\quad \quad \text{ for } \text{ some }\quad n\in ]i,j]\cap \mathbb {N}, \end{aligned}\) then we term it an \(\varepsilon \) -affine sequence. Such a sequence can be decomposed as the algebraic sum of an affine and a bounded sequence whose supremum norm does not exceed \(\varepsilon .\) Also, it is interesting to observe how convexity behaves differently for functions and sequences highlighting the contrasts between the continuous and discrete settings.
