Prove that \(\sum\limits^{\infty}_{k=1} O(f_k(i)) = O\left(\sum\limits^{n}_{k=1} f_k(i) \right)\) by using the linearity property of summations.

Let \(g_1, g_2, \dots, g_n\) be any functions such that \(g_k(i) = O(f_k(i))\). By definition, there exist constants \(c_1, c_2, \dots, c_n \ni g_k(i) \leq c_k f_k (i)\). Take the maximum value of \(c_k\) where \(1 \leq k \leq n\). This gives us

\[\begin{split} \sum\limits^{n}_{k=1} g_k (i) & \leq \sum\limits^{n}_{k=1} c_k f_k (i) \\ & \leq c \sum\limits^{n}_{k=1} f_k(i) \\ &= O \left( \sum\limits^{n}_{k=1} f_k (i) \right) \end{split}\]