TADM2E 2.13
From Algorithm Wiki
By Definition of Big-O we note that $ f_1(x) + f_2(x) = O(g_1(x)) + O(g_2(x)) $ and that this is equivalent to, $ f_1(x) + f_2(x) = O(max(g_1(x)) + O(g_2(x))) $ which is equivalent to,
$ O(max(g_1(x)) + O(g_2(x))) = O(g_1(x)+g_2(x)) $ QED.
