> [!NOTE] Theorem > Let $f_{1},f_{2},g_{1},g_{2}:\mathbb{N}\to \mathbb{R}$ be [[Real sequences|real sequences]] such that $f_{1},f_{2}$ are [[Big O Relation on Real Sequences|big O]] of $g_{1},g_{2}$ respectively, i.e $f_{1}=\mathcal{O}(g_{1})$ and $f_{2}=\mathcal{O}(g_{2}).$ If $g_{2}=\mathcal{O}(g_{1})$ then $f_{1}+f_{2}=\mathcal{O}(g_{1}).$ **Notation**: we may write instead if $g_{2}=\mathcal{O}(g_{1})$ then $\mathcal{O}(g_{1})+\mathcal{O}(g_{2})=\mathcal{O}(g_{1})$ which denotes for all $f_{1}=\mathcal{O}(g_{1})$ and $f_{2}=\mathcal{O}(g_{2}),$ $f_{1}+f_{2} = \mathcal{O}(g_{1}).$ **Proof**: ...