TADM2E 2.15

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Choose $ c_1 $ to satisfy $ f_1(n) \le {c_1}{g_1(n)} $ for all $ n \gt n_{1,0} $ and $ c_2 $ to satisfy $ f_2(n) \le {c_2}{g_2(n)} $ for all $ n \gt n_{2,0} $.


Note that $ c_1 $ and $ c_2 $ above can be substituted with $ c_3 $ such that $ c_3 = max(c_1, c_2) $ and the conditions still hold.


Since $ a \le b, c \le d \implies a + b \le c + d $ it follows that $ f_1(n) \le {c_3}{g_1(n)}, f_1(n) \le {c_3}{g_1(n)} \implies f_1(n) + f_2(n) \le {c_3}({g_1(n)} + {g_2(n)}) $.


Therefore $ f_1(n) + f_2(n) = O(g_1(n) +g_2(n)) $